A Security Deposit equivalent to 3.5% down on an FHA loan would allow you to fire your landlord and own your own home, at today's interest rates! Actually, 3.5% of a $600K purchase price would be $21,000 which is far less than the possible hefty pet deposit a landlord might impose. Your house payment may be somewhat higher but your two dogs will love you for it! You'll be able to deduct the interest payments, too!
Other than that option, you are must abide by the whim and rules of the landlord. There may be alternative properties in your price range that are 'pet friendly'... Good luck!