I recall something about getting Rex a real brain, but it was so long ago, I've forgotten. So, now I have a dog brain, but I don't know who to go to to have it transplanted. Also, this surgery won't change Rex's good qualities, will it?

go to lady Gibson. she will gibe a good brain for attack.

Go to the King in Free side, after you finish his quests talk to him about Rex. Then it will trigger "Nothing but a hound dog".

Jacobstown doc will do transplant

Options for brains:

Violetta, fiend dog, fed Turbo and the brain is bathed in the chemical still. Increases Rex's speed.
Rey (spanish for King, Rex is latin for King, I laughed): Gibson's dog, increases Rex's attack, he was a junkyard dog.
Lupa: Legion dog, increases Rex's defense (didn't pick it up, don't know why it increases def).

Doctor Henry in Jacobstown will perform the operation, Julie Farkas from the followers directs you to him after the King tells you to talk to her.