I'd say the difficulty you may face will depend on a) your budget, and b) how big the dog is.
When my wife and I were looking for a 1 bedroom place in London, we narrowed it down to a few neighborhoods (much like you have), and then just walked in to estate agents in each neighborhood and asked what they had in our price range. Usually they had at least 3 or 4 rentals to look at, and after maybe 2 or 3 days of looking we were able to find one we liked. Given that experience, I wouldn't suggest paying a relocation service.
I'd suggest having a look on rightmove.co.uk to see what's available in the neighborhoods you're interested in, so you can see how they line up to your budget. At least that might help you narrow it down so you can do a more efficient search when you come over.