Hi there, thanks for your question. Whereabouts is your puppy scratching/biting the most? This information can help narrow down possibilities. The most likely diagnoses for why your puppy is so itchy is mange (mite infestation), fleas and environmental allergy. There are 2 types of mites- one contagious and one that is already in the dog's skin but can flare up in times of stress, however minor the stress (eg. anaesthesia/surgery). I understand you cannot see fleas on your pup but this is still a possibility, especially if he is not on any prescription flea control. Some dogs are allergic to flea saliva so even one flea bite can trigger a reaction. He may also have an allergy to something in the environment- house dust-mites, grass, etc. Your vet can check for mites with a skin scraping and a blood test. Also I would advise putting him on a good veterinary flea treatment regularily, and obtaining a gentle shampoo from the vet. Also think about whether he has had any products applied lately (shampoos, flea treament) that may have triggered a reaction.
As for taking him to the vet, it may be something that you just need to do in this situation to resolve the problem. He can be sedated for the skin scraping. If your vet practice doesn't mind, you can take him in a couple of times and sit in the waiting room with him for a few mins without actually seeing the vet to make him understand that it's not always bad!! Please bear in mind that if he does have mites or fleas, these are contagious (to both dogs and people in the case of mites) so please keep him away from other dogs until confirmed otherwise, and please practice good hygeine when you are handling him. Hope this helps and please do not hesitate to reply with more questions.