For about $500 a month, you can get a decent townhouse with 30-yr mortgage at 110% financed as first time homebuyer for extra loot (furniture, fence, etc). Definitely the way to go if you have a large dog.
There are a few apartments in the Far West End area near me that allow pets, but those are usually restricted in size to 40 pounds. Have you checked the free Apartment Guide (at any grocery store) for their checklist of commercial places that allow pets?
If you want a private (non-commercially owned) place, look in the classified for ones that list a fence in the ad as they will be more likely to allow large dogs.