To be completely honest, I think the cold has to pass, teething pain under control, and you and hubby agreeing to try/do the exact same thing for it to work. If poor thing is upset to the point of vomiting, then you really have to regain his trust or else it will be much harder to sleep train. He will resist more bc of fear of being left alone.
For us, we EBF and at that point still had 2 NFs. It was probably habitual and just like your LO, he had to be held. It took about a solid 5 days for him to get the hang of pu/pd, but bc he would get hysterical, I stayed on the floor for another week or so and slowly phased that out. All in all it probably a few weeks but when he slept on his own, it was wonderful. There was crying involved during the process but I never left his side.
I think you have the right idea to ride the cold out, and get teething pain under control. We usually use motrin at night when I see inflammed gums and whites poking under gums. And another dose 6 hrs later.
Talk to husband about it and set a date to start after his cold resolves. If he's not on board even then, I suggest you get well rested right before and treck through it. Let me know what you think and we're definitely here to help you through this. You're not alone.