Bank site not fully opening on Macbook

Please tell us exactly what year MacBook Pro this is and exactly which macOS version it is running.

If you are running an older macOS, especially Catalina, then you really should consider upgrading the OS to something newer.

For example, a 2017 MBP can run macOS 13 Ventura, one of the three most current macOS versions and still well supported. Catalina and the older browsers that will run in it are shoeing their limitations on the modern, secure internet.

Meanwhile, you can try running Firefox if Safari cannot make secure connection to your bank's website. I believe the current secure version of Firefox would be the Extended Support Release (ESR) version 115. Mozilla is dropping support for Catalina so there will likely be no further updates.

Even If your macOS is in the last three supported versions, A bank refusing to quickly connect additional devices attempting to connect seems perfectly normal to me.

Even when I try to connect to the same banking sites with the exact same computer running the exact same MacOS, I sometimes get challenged and told to provide All identifying credentials anew. if I change ANYTHING, including MacOS version, I have to fully authenticate that computer again.

Your description does not sound like an exceptional problem to me. it sounds like your bank is being very cautious, which might not be such a bad thing.