How to exempt a banking website from HTTPS scanning globally at Avast s.r.o.?

Since Avast 2016, Web Shield supports “transparent” HTTPS scanning for certain browsers, including Firefox. In this mode, browsers see the original certificate instead of Avast’s one. See below for quoted release notes of Avast 2016:

If for some reason this “transparent” mode doesn’t work, then Web Shield will change the method to the old one: replacing the certificates (MITM). That’s what you’re seeing intermittently.

So basically, Avast is working as it should: changing scanning method to the old one to keep you safe. I don’t know why the first method sometimes fails for you, maybe a conflict with browser extensions or other resident software, browser configurations etc.