Well, like you said: Mouse cheats. Could well be, that he declared the world his temple and he took physical immunity or something.
But that doesn't really help your game, does it? Ok, back on track
You could go with a clause that he only has his toughness and recovery powers when acting to defend what (or who) he is sworn to protect. Preemptive strikes? No superdog. Sort of like how the Righteousness bonus works. Additionally, the toughness and recovery powers could be attached to a sponsor, who catches additional blows, but for a price in debt. Maybe it's a bit like soulfire, and if the temple dog uses his toughness powers, he needs to recuperate his soul at one point, so if he uses it too long, it comes at the cost of the destruction of his soul.
I know, those are all not really standard catches, but temple dogs are by all accounts anything but standard. But all of those can work to limit the temple dogs access to his power, if you want to, even if it is just by compelling him to do so.
Bear in mind, it's totally possible to have "Unknown [+0]" as a Catch. I mean, take, say, an elephant. That's at least Inhuman Toughness. But there isn't anything that would render an elephant as easy to kill as a regular human that would not work on an animal of similar toughness. Yes, you can kill an elephant with a lot of weapons, but elephant guns also do just as much damage to bears, or to people.
That's what "Hulking size" is for, it adds a few more boxes to your stress track, like toughness does, only without the armor.