The Residence Inn (a Mariott property) on Biltmore Avenue allows pets. It would be closer to the downtown/Biltmore area than say lakes. Nothing shy of a cabin or campsites are really near to the bigger lakes (Lake James/Fontana Lake). Now Lake Lure down in Rutherford County has various mom & pop type rentals - not sure on the pet policies.
If you're looking to come up here during the month of October, you better make reservations soon (if you haven't already). That is a time of year when hotel rooms become hard to find due to all the "leaf lookers".
Asheville is a wonderful place to visit, although, be prepared for a very long drive from Florida