Here is my experience.
My husband and I ate there last year. When ordering I told the waitress no beans, she nicely said those are the best part, I advised I couldn't eat them due to an allergy. (This is a non life threatening allergy, I just can't digest them properly. I safely eat bean-less Mexican all the time, including Chipoltle where the beans could easily cross-contaminate. I also could have easily just left them on the plate, but don't like to waste food).
Suddenly, she went into full "hovering mother mode" and verified that every ingredient in everything else in my dinner selection was okay for me to eat. She took my husband's order, and about 3 minutes later a manager came to our table. He did a second verification of my meal.
So, I don't know if this is their normal routine, or if I just happened to have any overly cautious waitress. I certainly did appreciate her concern, but for me in particular it was a bit over-zealous (not that she had any way to know that). When the food arrived, she confirmed everything on my plate, brought me new flatware, and a minute behind her was the manager again confirming my food was bean-free. My husband and I were chuckling about our VIP treatment.
I also don't eat tomatoes (for the same reason, but my meal did not include any) she noticed I was not using the salsa for my chips and offered to bring me guacamole. I declined, because I am used to eating naked chips; but we really did feel she went way above and beyond. On our way out that night we left her a compliment with Guest Relations along with a healthy tip, but I don't remember her name or I'd tell you.
If you really want to eat at the restaurant and your child is milk, egg, and peanut free. I think would be easy to accomplish. I just looked at the menu, and didn't see any entrees including peanuts. The easiest thing would be to get him a meat of some kind chicken or steak and a safe shell, probably corn tacos, (I don't know what is in a tortilla other than flour) vegetables, and make him fajitas. There are other options, but I am trying to think of kid-friendly non-cheese Mexican foods.
Also my best friend has a child with multiple life-threatening food allergies and they travel internationally, she makes flash cards for every language with the names of the foods her daughter can not eat and includes words like allergy and danger. If you are really worried about the language barrier, you could do the same. In my experience, both the people who checked my order had perfect English and barely any accent. BTW, since my friend also carries EPI pens through international airports, she also carries flash cards explaining why she needs to keep them.