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.
If it works on chromium I’d consider that even if it is a quirk on the bank website, chromium is handling it cleanly and allowing you to use the site. That’s something we probably want incorporated in Firefox. I’d encourage submitting the bug report to Mozilla, and don’t assume too much about what they can/cannot do!