You might want to take a look at the BankID app, which doesn't depend on functionality in the SIM card. Link in Norwegian: https://www.bankid.no/privat/bankid-app/
It doesn't work with my bank unfortunately, and it does not seem to replace "BankID on mobile".
Oh, thought all the banks supported that :)
If you get another bank to issue BankID to you, then you may be able to ask your current bank that you may log in using that BankID. And this way you may perhaps be able to use the app instead. Maybe.