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There may be a type where you compare $date1 to $date2. You typed $tdate1 < $date2 but $tdate1 doesn't exists. Try to fix that and see if it works.

Meanwhile I try to reproduce your use case in a dev environment to help you out. Feel free to tell me if it is ok for you.

Hope it helps. Take care.

UPDATE 1: Eureka! This is it! Basically you needed to use onContentAfterSave rather then onContentBeforeSave and use the FieldsHelper in conjuction with the FieldsModelField. Here is the code:

<?php
 
defined( '_JEXEC' ) or die( 'Restricted access' );

class plgContentAlertcustomfieldhandler extends JPlugin
{
   public function onContentAfterSave($context, &$article, $isNew)
   {

    if ($context === 'com_content.article') {
        $custom_fields = FieldsHelper::getFields($context, $article, true);
        $custom_fields_by_name = \Joomla\Utilities\ArrayHelper::pivot($custom_fields, 'name');

        $date1 = strtotime($custom_fields_by_name['date1']->value);
        $date2 = strtotime($custom_fields_by_name['date2']->value);

        if ($date1 < $date2) {
            //define the value you want to assign
            $your_custom_field_value = 'alert';

            JModelLegacy::addIncludePath(JPATH_ADMINISTRATOR . '/components/com_fields/models');
            /**
             * @var FieldsModelField $model_field
             */
            $model_field = JModelLegacy::getInstance('Field', 'FieldsModel', ['ignore_request' => true]);

            //set the value using field model instead to make change permanent in db
            $model_field->setFieldValue(
                $custom_fields_by_name['alert']->id,
                $article->id,
                $your_custom_field_value
            );
        }
    }

     return true;
     }
  }

Have a great day. Happy coding!

UPDATE 2: Second version of the code toggling on and off the checkbox when your requirement is not met.

<?php
 
defined( '_JEXEC' ) or die( 'Restricted access' );

class plgContentAlertcustomfieldhandler extends JPlugin
{
    public function onContentAfterSave($context, &$article, $isNew)
    {

    if ($context === 'com_content.article') {
        $custom_fields = FieldsHelper::getFields($context, $article, true);
        $custom_fields_by_name = \Joomla\Utilities\ArrayHelper::pivot($custom_fields, 'name');

        $date1 = strtotime($custom_fields_by_name['date1']->value);
        $date2 = strtotime($custom_fields_by_name['date2']->value);

        $your_custom_field_value = '';
        if ($date1 < $date2) {
            //define the value you want to assign
            $your_custom_field_value = 'alert';
        }
        JModelLegacy::addIncludePath(JPATH_ADMINISTRATOR . '/components/com_fields/models');
        /**
         * @var FieldsModelField $model_field
         */
        $model_field = JModelLegacy::getInstance('Field', 'FieldsModel', ['ignore_request' => true]);

        //set the value using field model instead to make change permanent in db
        $model_field->setFieldValue(
            $custom_fields_by_name['alert']->id,
            $article->id,
            $your_custom_field_value
        );
      }

      return true;
     }
 }

Basically you needed to use onContentAfterSave rather then onContentBeforeSave and use the FieldsHelper in conjuction with the FieldsModelField. 

Here is the code:

<?php
defined( '_JEXEC' ) or die( 'Restricted access' );

class plgContentAlertcustomfieldhandler extends JPlugin
{
   public function onContentAfterSave($context, &$article, $isNew)
   {

    if ($context === 'com_content.article') {
        $custom_fields = FieldsHelper::getFields($context, $article, true);
        $custom_fields_by_name = \Joomla\Utilities\ArrayHelper::pivot($custom_fields, 'name');

        $date1 = strtotime($custom_fields_by_name['date1']->value);
        $date2 = strtotime($custom_fields_by_name['date2']->value);

        if ($date1 < $date2) {
            //define the value you want to assign
            $your_custom_field_value = 'alert';

            JModelLegacy::addIncludePath(JPATH_ADMINISTRATOR . '/components/com_fields/models');
            /**
             * @var FieldsModelField $model_field
             */
            $model_field = JModelLegacy::getInstance('Field', 'FieldsModel', ['ignore_request' => true]);

            //set the value using field model instead to make change permanent in db
            $model_field->setFieldValue(
                $custom_fields_by_name['alert']->id,
                $article->id,
                $your_custom_field_value
            );
        }
    }

     return true;
     }
  }

UPDATE:

Second version of the code toggling on and off the checkbox when your requirement is not met.

<?php
defined( '_JEXEC' ) or die( 'Restricted access' );

class plgContentAlertcustomfieldhandler extends JPlugin
{
    public function onContentAfterSave($context, &$article, $isNew)
    {

    if ($context === 'com_content.article') {
        $custom_fields = FieldsHelper::getFields($context, $article, true);
        $custom_fields_by_name = \Joomla\Utilities\ArrayHelper::pivot($custom_fields, 'name');

        $date1 = strtotime($custom_fields_by_name['date1']->value);
        $date2 = strtotime($custom_fields_by_name['date2']->value);

        $your_custom_field_value = '';
        if ($date1 < $date2) {
            //define the value you want to assign
            $your_custom_field_value = 'alert';
        }
        JModelLegacy::addIncludePath(JPATH_ADMINISTRATOR . '/components/com_fields/models');
        /**
         * @var FieldsModelField $model_field
         */
        $model_field = JModelLegacy::getInstance('Field', 'FieldsModel', ['ignore_request' => true]);

        //set the value using field model instead to make change permanent in db
        $model_field->setFieldValue(
            $custom_fields_by_name['alert']->id,
            $article->id,
            $your_custom_field_value
        );
      }

      return true;
     }
 }

UPDATE 2: Second version of the code toggling on and off the checkbox when your requirement is not met.

<?php

defined( '_JEXEC' ) or die( 'Restricted access' );

class plgContentAlertcustomfieldhandler extends JPlugin
{
    public function onContentAfterSave($context, &$article, $isNew)
    {

    if ($context === 'com_content.article') {
        $custom_fields = FieldsHelper::getFields($context, $article, true);
        $custom_fields_by_name = \Joomla\Utilities\ArrayHelper::pivot($custom_fields, 'name');

        $date1 = strtotime($custom_fields_by_name['date1']->value);
        $date2 = strtotime($custom_fields_by_name['date2']->value);

        $your_custom_field_value = '';
        if ($date1 < $date2) {
            //define the value you want to assign
            $your_custom_field_value = 'alert';
        }
        JModelLegacy::addIncludePath(JPATH_ADMINISTRATOR . '/components/com_fields/models');
        /**
         * @var FieldsModelField $model_field
         */
        $model_field = JModelLegacy::getInstance('Field', 'FieldsModel', ['ignore_request' => true]);

        //set the value using field model instead to make change permanent in db
        $model_field->setFieldValue(
            $custom_fields_by_name['alert']->id,
            $article->id,
            $your_custom_field_value
        );
      }

      return true;
     }
 }

There may be a type where you compare $date1 to $date2. You typed $tdate1 < $date2 but $tdate1 doesn't exists. Try to fix that and see if it works.

Meanwhile I try to reproduce your use case in a dev environment to help you out. Feel free to tell me if it is ok for you.

Hope it helps. Take care.

There may be a type where you compare $date1 to $date2. You typed $tdate1 < $date2 but $tdate1 doesn't exists. Try to fix that and see if it works.

Meanwhile I try to reproduce your use case in a dev environment to help you out. Feel free to tell me if it is ok for you.

Hope it helps. Take care.

UPDATE 1: Eureka! This is it! Basically you needed to use onContentAfterSave rather then onContentBeforeSave and use the FieldsHelper in conjuction with the FieldsModelField. Here is the code:

<?php

defined( '_JEXEC' ) or die( 'Restricted access' );

class plgContentAlertcustomfieldhandler extends JPlugin
{
   public function onContentAfterSave($context, &$article, $isNew)
   {

    if ($context === 'com_content.article') {
        $custom_fields = FieldsHelper::getFields($context, $article, true);
        $custom_fields_by_name = \Joomla\Utilities\ArrayHelper::pivot($custom_fields, 'name');

        $date1 = strtotime($custom_fields_by_name['date1']->value);
        $date2 = strtotime($custom_fields_by_name['date2']->value);

        if ($date1 < $date2) {
            //define the value you want to assign
            $your_custom_field_value = 'alert';

            JModelLegacy::addIncludePath(JPATH_ADMINISTRATOR . '/components/com_fields/models');
            /**
             * @var FieldsModelField $model_field
             */
            $model_field = JModelLegacy::getInstance('Field', 'FieldsModel', ['ignore_request' => true]);

            //set the value using field model instead to make change permanent in db
            $model_field->setFieldValue(
                $custom_fields_by_name['alert']->id,
                $article->id,
                $your_custom_field_value
            );
        }
    }

     return true;
     }
  }

Have a great day. Happy coding!