First, I think you'll find your doctor is right. I'm not sure you even need a prescription for those drugs in Mexico. Most everything is OTC. As long as you're not planning on bringing the drugs back over the border, you should be OK.
Even then, if you put the drugs in your prescription bottles you would probably be OK, though I wouldn't try it. They really frown on trying to smuggle drugs into the States from Mexico, even small amounts. And your big problem is the dogs they have there. They're trained to smell narcotics. They don't care about narcotics going INTO Mexico. But they sure do care about narcotics coming out.
Well, that's neither here nor there. You're going in and staying, so you should be OK.
Healing hugs and lots of love,
Dan