I could see maybe allowing an app on a locked down tablet provided by the school and wiped before the exam. Still a big hardware and IT burden. And you need to lock the student in the app, not record if they close it, because it's so easy to accidentally switch apps with a stray finger or misplaced muscle memory.
Bring your own device sounds like a mess. Someone would create an app that looks like the locked down app but isn't really, etc. Or students would sneak in a second phone.
You really don't want to give teenagers the temptation to sneak online or into their notes during exams.