Should dog be euthanized if he bites?

We adopted a lab / hound mix dog from a local shelter. They didn't know much about his history, just that he he been found living on the streets. He seems to be about 2-3 years old. He has several scars and bot his ears are torn. The shelter did behavior testing on him to see how he would react if food was taken away from him and he "passed". We got him about 6 weeks ago and he seems "scared" sometimes and cowers when you go to pet him. He loves people, though and very much wants our attention. He has been to the vet, to a groomer and to doggy day care, and been around many types of people with no problems. But, last Saturday, he was asleep in the laundry room and my husband opened the door to let him out. He was leaning against the door so he had to push a little harder to get him to move and when my husband put his hand it to pet him, he bit him. It broke the skin and bled. We contacted a behaviorist / trainer and she has suggested using positive reinforcement. If a dog bites in this situation, does it need to be euthanized? She alarmed us a bit by saying that if he does it again he might not be able to stay in our home or any other home. He has snapped one other time when he was sleepy and resting. Otherwise we have not seen any aggressive behavior around food or in other circumstances. He approaches people in a friendly way and wags his tail. Still, the trainer said if he bit once he will likely do it again. She said to walk by him when he is resting or sleeping and throw treats at him so he learns to associate those situations with something positive. In your experience, can a dog that bites in this kind of situation be rehabilitated?
Thank you!