A root canal at 4?? No chance I would let my son have this done without a 2nd opinion. First of all, I think you should go to a pediatric dentist if you aren't already. I think the term "family dentistry" is pretty loose and really is geared more toward adults.
I'd check out pediatric dentists in your area (check reviews to find the best one), and call them. Explain your situation and ask whether your current dentist can fax over x-rays for them to take a look at and give a 2nd opinion. Then if he does actually need a root canal and it's legit, you're dealing with someone 100% trained in administering anesthetic to children and also possessing the bedside manner for children.
At our pediatric dentist, the appointment also costs around $300 - but that was just the first one. I was reimbursed for most of it later from my insurance co. Now the dentist just estimates my bill and charges me that...if there needs to be an adjustment, they either send me a check or a bill. I just took in my 5 year-old and 2 year-old and our whole visit cost $60. And they did x-rays on my 5 year-old as well as a check-up, and a first-time check-up on my 2 year-old.