I just got a very good condition P345 at my LGS for $200.
The owner is a friend and it’s been there awhile, so he gave me a discount.
I’m a fan of the P series and know about the potential firing pin damage to the 345 if it was dry fired much without a mag inserted.
Is there any way I can tell if the damage has already occurred?
If I fired a few hundred rounds without a failure to fire should I be GTG?

You should be GTG or you can remove the Magazine disconnect to prevent the damage.