Having just looked at homes.... the first place wanted to charge an extra $50 per dog per month on top of a rather large pet deposit. Bwwhhaha. NO.
The place we have chosen, however, is still the same amount per month with an even larger pet deposit as the house is newly remodeled and they are concerned with wear and tear- which is completely understandable.
I would never pay an additional fee per month, that is what the deposit is for. Maybe offer a larger deposit, you know Sawyer is well behaved and will not cause any damage, so when you move out, you know you'll get it back. No worries there. If you don't like it, find another place to take your money.
I've found if you are up front and honest it always works better. When I emailed around to these places, the very first thing I said is, "we have three big dogs and are willing to pay an extra deposit. will you accept them?" Every place that got back to me said they would. You just gotta take control of the situation.
Good luck.