I have allways been concerned with the scenerio of flying a dog (pet/family member) in the lowers as you have no idea if the flight crew will monitor temperature or give a damb. Not to mention on many fleet types lower bay temps on long flights are hard to keep warm.
If you can bring aboard on an AMC flight that would be great, I do not expect there would be quarintine issues between the us and uk, best check on it though.