Now that MacOS Catalina has moved to zsh, can I upgrade bash 3.2.57 with 5.0.3? I understand apple's move from bash to zsh, though I'd really rather that they hadn't made the change.

I've tried zsh, and decided that I'd prefer to remain with bash.

So, since MacOS Catalina isn't using /bin/bash any more, am I going to run into any problems if I replace it with the new version? (  3.2.57 with 5.0.3 ).

I wouldn't have thought of doing this before, because it would have been quite possible that some part of the OS relied on a feature that had changed in new versions.

Now, though, that /bin/bash isn't officially used, can we change it?

If so, that would be a really positive side-effect of the move to zsh. We can keep our user accounts compatible with out linux accounts, all using zsh, and keep the latest version running on all our machines, without having any danger of a conflict with the OS.
You can't easily (and shouldn't) change/upgrade /bin/bash. You can install a more recent version of bash through Homebrew though, and use the information from How to use bash as default shell to change your login shell.

This will not break any existing bash scripts as these refer to #!/bin/bash in the first line.