To me, for a pet photo with Santa, 'Santa' is just a prop like a giant candy cane or a reindeer. As long as the Santa cues are there - white beard, red suit/hat, boots - I don't think the ethnicity or gender of the person playing Santa would matter at all. The only thing visible in traditional Santa-photo gear is the eyes, nose, and a bit of the cheeks, as Sio illustrated with her photo above, so I can't see it being a big deal at all.
In fact, the dog rescue group I'm associated with does Santa photos as a fund raiser, and we're almost exclusively female, so all our Santas are female. Doesn't bother the dogs any :-) and as far as I know, we've never had any negative feedback.