That's what makes heating pad brooding such a departure from conventional heat lamp brooding. You want the place you're brooding to be as cool as possible, outdoors even. The only warm spot is directly under the heating pad. And it's not supposed to be 95* inside the cave. That's not how it works. You're not trying to heat up the whole inside. It's the direct contact with the pad, mimicking snuggling up against a broody hen, that warms the chicks.
Right. Take the thermometers out and stick the back of your hand against the pad inside the cave. That is what the chicks will be doing to warm up. And a pile of little birds is pretty warm all by itself.