I think the key part here is that it’s a guess on your part whether using Firefox is the cause. Do you get any specific error when using the website? Or does something just “not work”, such as you click a button and it does nothing?
Also, I’ve run into stuff like this before, and my best bet has been to be flexible about using other browsers to work around issues. I would suggest testing the banking website with Chromium (or even Chrome). If it works, file a bug with Mozilla (https://support.mozilla.org/en-US/kb/file-bug-report-or-feature-request-mozilla) and just use Chromium/Chrome for only that website until the bug is fixed.
This will allow you to still do business, while still participating in open source via a helpful bug report that could end up benefitting others as well.