Might want to add a query string parameter to all gui files like 

(URL).main.css?v=1.0.1

where you can auto generate version number from assembly info. If you use css or js bundling and minification they usually do something similar to avoid cache problems (they usually append a hashed key of the content so if the content of the css changes the hashkey in the URL will be changed too and the client will be forced to request the file again). So adding that will both improve performance and solve your caching problems.

Adding support for bundling and minification is so simple today so I would recommend that route. It's a few hours of work and then you get improved performance on the entire website "for free".

You can use ETags for the DXC to more granualarly control when the CDN cache is cleared thereforce allowing a long cache header but an ETag to force an update when you need. Possibly via code or adding a version config attribute/admin area.

Or Like Daniel said add querystrings, also you could bundle your assets together so they have a unique URL, that's what we do and it works.