Do the popups contain any information that would be interesting for search engines to crawl? If not, I wouldn’t worry about it. If yes, maybe look into alternatives for how you code that.
However switching from hand rolled to WP “because SEO” seems a bit of a cheap-shot to me. It’s not like WP will magically give you more SEO juice than other platforms. Sure, they have some optimizations, but you can code those into your hand rolled solution too.
If you are able to write code yourself, you only realy need a CMS if you have a client (who can’t code) that needs to edit their content.
The chances are that moving from hand-coded to something like Wordpress will only slow your site down with bloat, which can be to the detriment of SEO.
Ask this someone how Wordpress will handle the pop-ups in a more SEO friendly way.
Then you could consider implementing these methods in your own code.
Though I suspect (having made such a suggestion) they won’t actually know.
Why not make a URL for the article as well and then link to the article from the button, and then with javascript prevent that and open a modal instead. That way search engines can go to that URL and read the article contents.
This will also ensure that people with javascript disabled also have a way to read your articles.