how to help if a dog is afraid but is the only that you care for that acts this way?

The dog is going to have to get used to you, which means you have to spend time at the client's home. You won't be able to establish any level of trust with the dog if you go in, do the feeding and walking and leave. I'm afraid you would need to be around the dog for hours. I had two chihuahua mixes staying with me. One was completely docile; the other had attitude and was protective of her sibling. Each wore a harness. There was no way I could put the harness on the non-docile one, without her trying to bite me. I could attach a leash to her collar, but that wasn't a safe way to walk her because of her head/neck and she could easily get away. It took nearly two days for that dog to trust and like me. In fact, the owner was quite surprised by the relationship we had established after a week. The dog was afraid and only one-on-one time with the dog can overcome that kind of fear. Being with the dog for a half hour two to three times a day probably won't resolve his issues.

Has the Pet Parent given you any information about the dog that is so frightened? How does he get along with the other dog? Observe their interaction. I would sit down on the floor beside the frightened dog, caress him and speak very softly. Put on some soft music like Sound Scapes or what massage therapists use for relaxation. It will take time to gain his trust and develop a bond. Find the "spots" he likes to be rubbed or scratched i.e. the ears, base of tail, belly rubs, etc. Each dog has their favorite spots. Move slowly but smoothly around him. Ask the Pet Parents for more background information and suggestions.

I would read some articles on credible websites on dog training and anxiety. I think the first thing to try if you can get him moving without dragging him is exercise. Don't bother with food since he's clearly too anxious to eat, and this can undermine your efforts if you keep trying. Don't use much sound or excitement, as this can make it worse. Above all, just be calm. If he specifically doesn't like the leash, I would practice everything leading up the the leash (calling him, etc) without putting the leash on, teaching him that just because certain things are happening doesn't mean the leash will go on. I would just get him used to being around the leash without putting it on, and when you do put it on, take it off immediately just to show him it doesn't have to stay on long and is nothing to be afraid of.