Was thinking about this last night and I think one of the big things people are forgetting is that the OP was found by Agency A and not Agency B. Those suggesting that they should just get the same person (or using an analogy of the same product) from Agency B forget that they can’t, only Agency A contacted him, got him to interview, got him through the process.
This is why Agency A should get paid, they used their expertise to find a candidate, contact him and market him to a company who they knew would like him, they did this so well that they got an interview and then a placement out of it. The OP got a job he wanted at a rate he was happy with, Agency A got a fee, the client got the right man, all was well in the world.