I've been running a 35" tire on my stock tailgate for a month. It fits perfectly against the cutout in the the stock bumper. However, if the backspacing were 0.5" more or less, this might be a different story.
The weak point seems to be the tailgate hinge, not the plastic mount, so the reinforcement wouldn't help here.
I'd go for a replacement tire carrier like the one made by Teraflex, or get a bumper with a swing out tire carrier. I'm waiting on my ACE rear bumper / tire carrier, but I haven't had any problems with the stock setup.