The problem is with the Banking apps. They would not work on any ROM which does not have the full compliment of Google’s files and infrastructure to help it. There may be some exceptions to this rule. Some apps that may work. You can check with the developers - write to them and ask them to support /e/ or non google ROM’s. One option to install Banking apps on /e/ was discussed here. Alternately some banks have PWA which work on /e/ or other custom ROM’s.
On a personal note I never installed any banking app on my phone even when using stock ROM so not exactly an expert to speak on this subject.
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