Sorry to hear you're struggling. If you have plateaued in your treatment, it might be time to switch meds. For most of my treatment so far, I was on two meds at a time. My doc would switch them out when my improvement would slow or stop. I am finally down to one abx now after 13 months, a high dose amoxicillin. It sounds like you're doing the right thing in going to a new doc if your current one isn't doing the job.
Regarding your son's vaccinations, I think I would make the same decision you have. Assuming that he's not breast feeding, he's not getting your natural immunity. Taking public transport in a big, international city is a very good reason to get him vaccinated.
What I did with my daughter is to spread them out more. I just didn't want to overload her immune system at one time. The schedule here has babies getting up to five vaccinations at a time. I told the pediatrician I wanted no more than 2 or 3 at once, and then I'd come back a week or two later for the rest. The doc had no problem with that, but I had to remind him each time. Even if your pediatrician frowns on this or lectures you about it, just remind him or her that you will indeed vaccinate, but not all at once.
As long as Connor is doing well, don't worry too much about the "what ifs". (Kudos to you for getting in touch with Dr. Jones!) Babies are truly little sponges. He will pick up on your fear. Focus on getting yourself well. If he does develop symptoms later, you'll need to be well yourself so you can be ready to tackle that. It sounds like he is in great shape and you have good reason to be optimistic.
I am so sorry for your stress and concern. Living in the UK, I am not familiar with the medical system there and how they treat Lyme. I also don't know anything about treating infants.
If your naturopath can suggest immune system supplements for your infant, that is where I would start. You could also research the subject online. Would his pediatrician have anything to boost his immune system?
In your case your case, it is normal to feel stressed for your child Anxiety is a symptom in Lyme and especially its co-infection of Bartonella. I have had to take anxiety medications for my anxiety. You may do better getting something from your doctor to help you.
When things are not in our control, it is better to take things as they come and not worry but with lyme sometimes that is impossible. I had to take medication to help me do just that. My fiancé works for the government and has been away for over 7 months all over the world in dangerous places so since I had no control over it, the medication helped me deal with it.
I wish I had better and easier answers for you. Please keep us posted.