In the wild, chameleons eat all sorts of non-insect food matter. From birds to lizards, I've seen photos of everything. However, in the wild it's a necesity thing - you may not see anything really good in days, and if a nice little lizard crawls infront of you, you're going to take the chance because you might not eat again in a couple days. But in captivity, it's not really doing much since you can spice up the diet with so many other, more insectivore-specific bugs. And as Veiled is saying, the animal protein in mice is tough for insectivore animals to digest, since their systems just aren't specialized for that. And an excess of it will lead to medical issues.
It's a little like dogs. Some [insane] people think they can keep their dogs on a vegetarian diet, which cuts about 7 years off the life of your dog. Why? Because even though dogs are a tiny bit omnivorous and can happily tolerate some veggies and fruit, their digestive system and body is built for the digestion of red meat. So a carrot here and there is absolutely fine, but as soon as you make it a large chunk of the diet you're hurting the animal.
I think that perhaps if you really had to, a few a year wouldn't do much harm, but I just don't see any great points in favor of the practice. If you need extra fat, we have wax woms. If you need more calcium we have supplements and phoenix and butterworms. We have everything they want in insects.