You might need to give a more specific example of what the module area is doing for the ideal suggestion. However, a few ideas that might point you in the right direction:
Plugin
Create a plugin that runs before the "Content - Load Modules" plugin. You will need this to detect the page's view and, if it detects the blog category layout, strip any {loadposition}
that you do not want to display so that when "Content - Load Modules" runs, the unwanted modules no longer exist in the code it is parsing.
Use Read More
Set the category view to only show the content before the Read More break, and the article view to only show the content after the Read More break. Manually place the modules where they should appear on the category view, ie on one page it will appear both above and below the Read More. This could end up being quite fiddly for the admin who may have to duplicate a lot of content, depending on how your site works.
CSS
Hide the unwanted modules using CSS in the template. Not best practise, but should be a quick-and-easy approach.
Module
Update (if your own), or override (if core or third party) the template of the module so that it counts how many times it has been loaded on a page, and output content or output nothing accordingly.
Design
Does it need to appear in the article, or could you place the module elsewhere on the page without harming usability? If the admin is having to place the same module every time an article is created, it sounds like it may be a design problem.
Update
Is there any reason why you can't create a module area in your template - above or below the component area - and assign the module via the module manager? From the information you have given - this sounds like the best approach.