Hi--so sorry you're having this problem--I'll gladly give you some advice and information.
Yest, methotrexate gives tooth sensitivity an can cause the teeth to break. Is the infection you're suffering from an abscess in the gum or a large cavity? It is appropriate for the dentist to want to do a blood test before the procedure as the medications can lower your white blood count and limit your ability to fight infection. Good for him not just plowing ahead.
Still, this is a painful and distressing problem. I doubt it's just the RA but RA can cause thinner bones (compounded by the drugs for RA, especially steroids) so maybe it would also be a cause of teeth breaking.