First, Jogi, Are you looking for a furnished temp-rental apartment for all this time? Or a starter apartment to use for a few months while you look for something more permanent?
My Westie usually travels with me. Before I moved here in 2006, I visited frequently, and I always always found the temp apartment search to be a challenge, but not insurmountable.
First, tell the agent at the rental firm you want to use that you need dog-friendly accommodations, and consider offering double the usual security deposit, which the agency may request in any case.
The better firms will know which of their owners don't object to well-behaved pets. If the agent tells you no pets are allowed in any of their apartments, just contact a different agent. I know
www.apartmentsba.com offers some pet-friendly apartments (including one of mine that's unfortunately too small for your family). An excellent small firm that doesn't mind dogs is
BA Best Stay, whose owner is a doll and will help you find what you need.
If you're looking for a long-term rental, you have many more complications to resolve than just the dog. You can find a wealth of info on this forum to help overcome the difficulties foreigners face with long-term rentals. Still, owners will respond to cash, either in terms of increasing the rental fee a little or - my preference - offering a largish security deposit. If your dog is well-behaved, you'll get the deposit back. If he's not, the owner deserves some compensation for the extra wear and tear.
Best of luck! If you need help with the temp-rental agencies I mentioned, send me a PM.