Factual error: There is not a cat in hell's chance that Columbo would be allowed to take charge of the dogs Laurel and Hardy unsupervised and unaccompanied, regardless of his faith in their "deprogramming." These dogs are known to have killed a human being! If they had to be moved, they would be muzzled, attached to a very sturdy chain leash, and each would be led by an armed police officer who would have orders to shoot them if they showed any signs of hostility.
Revealing mistake: At the end when Laurel and Hardy are licking Columbo, a substance is caked on Columbo's neck, presumably the food they used to get the dogs to lick Peter Falk.
Answer: In the show, he was already a Lieutenant. A promotion would put him in the next rank up, which for the L.A.P.D. would be Captain. However, some of a Captain's duties would be overseeing other officers and ensuring they're compliant with policies, regulations, and standards. It would also most likely take him out of the field. This is something Columbo has no desire for as he rarely goes to police HQ's. Nor does he show interest in compliance and standards (for example, not going to his semi-annual evaluation at the firing range). However, he could still be assigned to a higher pay grade based on expertise, which is a form of promotion that does not include rank advancement. This would be going from Lieutenant I to Lieutenant II. I don't believe in the show it's ever started what his pay grade is. Although, in s02e01 (I believe) he mentions making $11K a year. Whether or not this was a true statement on his part, if you could find pay scale information for an LAPD Lieutenant in the 70's, it could give you an idea of his pay grade.
Bishop73