It sounds like Sid loves other dogs, so getting HIM a puppy may help his anxiety. He may bond to a new pup more than to you, Sid may just be your girlfriend's dog and the pup may be yours, who knows. Sometimes a dog (of any breed) will bond with one human more than another…and...like cats, they seem to sense when you REALLY want them to like you, and hold back!
I know of several single dogs who had issues who were helped by the addition of another basenji, it takes the pressure off them!
Good luck with Sid. If he was Vickie's dog, I'm sure he was well loved and "snatched" up from a wonderful life for one of uncertainty, so it may just take a long time to completely trust and relax.