I'm confused. The app's Automatic authentication option on a Mac has no bearing on your iPhone's lock screen time out. Those are separate things.
If you iPhone is locking immediately, then go to the iPhone Settings and change the timeout in Settings ➜ Face ID & Passcode ➜ Require Passcode, try something other than "Not Immediately".
Do note, On iOS 18, if you have Apple Pay enabled, or Face ID turned on, it will no longer have any other options besides "require immediately". But again this is not directly tied to the Mirroring App on the Mac at all.