Is it possible to read a custom, non-payment, ISO7816 AID on a multi-application smartcard that also has an active payment AID? I guess Apple Wallet may auto-detect the payment scheme and open, but this could be prevented if my app was already running in the foreground and scanning for the custom AID?
NFCTagReaderSession for non-payment AID on payment card
If you start your readerSession() before getting near the smartcard, this will prevent the "Wallet" app to be launched.
But, whether this kind of a mixed card will work without any issues remains to be seen, as things will depend on what AIDs the card will present, and whether the payment AID will cause an issue with your app at that point.
In any case, as long as the app has an active session beforehand (although keep in mind that sessions have a timeout, so you can't have one active all the time your app is in the foreground). That would be something you would have to consider when designing your UX flow.