Been thinking about this pup all day... Too many variables to assume this pup cannot be vaccinated. The pup is tiny.. less than 2 lbs, the DHPP was given every 2 weeks, the addition of Coronavirus to the final vac series, the vet did not administer epinephrine which is standard treatment for Severe Anaphylactic Shock.
Core vaccines can be given every 4 weeks: Two weeks after vaccination the body is at it's peak producing antibodies. To vaccinate a small puppy again at this time is overloading his system. Had he been vaccinated at 4 weeks, the immune response has tapered off to almost nothing and it would have been a safer option to vaccinate at that time.
There are many different manufacturing processes for the core vaccines. The pup's reaction could have been caused by any component in the vaccine used. The next step would be to use a different manufacturer for the core vaccine, and have a dose of epinephrine drawn up and ready, along with oxygen and fluids (also for any vaccines in the future). Consider pre vac treatment with Benedryl. And I don't mean to go do this now.
First, I would have the titers done, to see where his immunity level is at.
If he needs revaccination, don't do it soon. Wait, but keep him in a protected environment, let him grow up a bit. The bigger he is, the better he will be able to handle being vaccinated. If his titers are low or non existant, he will need one series, repeat at one year, then every 3 years or titer, as an adult to establish protection.
Rabies can be done later, at 8 - 12 months, same precautions as above. If he stays tiny, consider doing 1/2 vaccine (although illegal to do in my state, my goal would be to establish protection moreso then to be legal), then titer and repeat in 4 weeks (1/2vac) if needed, then titer and repeat at one year.
This would be my plan, if he were my dog, as opposed to being a 'bubble' dog. If at any time he experiences severe anaphylaxis (needing more than Benedryl to control reaction), scrap this plan, get the letter and protect him for life.
This is just an option for you to consider, discuss with your vet, get a second opinion on. There is risk involved either way, but if he can tolerate minimal vaccination, he can experience more things throughout his life, like going to parks, and stores, and meeting other dogs and having fun. Not that being a house dog wouldn't be fun too....