I once lived near a women that had a "pet" tiger. She was an eccentric women that lived on a small plot off land that was largely wooded. She was nice and the tiger appeared nice, she introduced me to it once.
What I didn't ever tell her was that I had my 375 loaded with 5 300 grain trophy bonded bear claws. I regret selling that gun.
That said the question I would ask would be what kind of shooting would you be doing in a honest scenario like that. It would likely be defensive or chasing a wounded (by a handgun most likely) game animal. So maybe a bolt action 375 wouldn't really be best? Perhaps then a ten gauge would be better, especially for a smaller large cat.
As far as anticipating escapes that would depend on your area, I don't anticipate any trouble from large dangerous game animals, however defense from pit-bulls and meth zombies is more likely.
