Hi there,
that's an awkward one, we too have a rescue dog although he is over his nervousness now!. How big is the dog, i.e if he's a small dog you could put his bed on top of the weighing scales and let him go to sleep. We also have two cats, one of which is absolutely terrified of the carrier I have to use to bring her anywhere so the vet suggested just leaving it in the room with the cat for a few days and maybe feeding her in it - worked really well, she totally relaxed and i was able to get her into it - again, feeding the dog on the scales might only work if he is quite small. Although, you could perhaps place a light sheet of wood over the top of it, reset to zero, this would give you more 'floor space' for the dog to stand on. All depends on size of dog really. Otherwise I would ask the veterinary nurse could you just pop in and weight him from time to time, surely they could let you do this without charging??
PS: Our dog is on a constant diet, I found that beta light is a good food for them, seems very filling.