Hi Mike,
Just to share our experience, we have kind of a similar flow where a prescriber invites a patient using his phone number, the patient receives an SMS, installs the app, and logs in using his phone number.
We've had to implement a intermediate API that communicates between our mobile and Medplum in order to store our credentials in a secure manner.
It actually helps us a lot to use the intermediate endpoint to handle strong or complex business logic.
Tell me what you decide to do, but I would definetely not store credentials in the mobile app itslef.