I flew my dog and cat via BA from DC to Heathrow. I think I paid $2,279 (it was definitely between $2,200 and $2,300).
I called about 50 people about properties and got a ton of "your cat is fine, I might consider a small dog, but not a big dog". I was looking at flatshares, so I don't know if it's easier without dealing with other tenants. Of the 50, I found 3 that were willing to consider my boys, and luckily my flatmate is awesome and really likes my pets, and my landlord's cool with it too. It took about a month to find a place.
I'm not sure what I'd recommend as far as paving the way...if you rent now, a letter from your landlord about you/your dog as good tenants might help. I think a lot of places just don't want to deal with the hassle of pets, and/or the noise of dogs.
Oh, and my boy's a lab/shepherd and is about 60lbs.