I'm not familiar with the specifics of iOS's interruption of other apps when calls come in, so there may be a simpler solution somewhere in the iOS device API.

An end-around solution might be to keep the call alive on a central server that acts as an intermediate switchboard (a third caller, sort of), so that even if one user disconnects, the call itself is still live. Then the app could attempt to rejoin the call once the interruption has passed. (Or, if rejoining is impossible, reinitiate the call.)

This makes several assumptions, so you'll need to dig around in the API documentation for both iOS and Twilio. However, assuming there's not an existing solution inside iOS, creating a call controller on an external server should be able to solve (or at least work around) the problem.

If you'd like to get a second opinion or a more in-depth review of options, I'd be happy to schedule a short call to review options, and — if necessary — some hands-on work with your code to help you solidify a fix to the problem.

Good luck!