but I'm logged into the front end, and am able to access Client Area, where billing and everything else is. 

When attempting to update to new version, system warns me about the unsecure connection (The information you’re about to submit is not secure), and when I insist on going further, nothing happens - I'm sent back to the login area. 

In the meantime, front end and Client Area are fine. 

The problem you have there is your server is forcing you to https, but you havent set the software to use it. You need to follow the instructions here to switch to https correctly

https://invisioncommunity.com/4guides/advanced-options/configuration-options/using-ssl-https-r273/