Hyde glue won't hold up -- as you see -- because of the heat. I'd be inclined to use something that's a gel that is catalyzed. I'd pick a gel because, once you clean both surfaces, you'll have space that needs to be filled with something. I'd use a catalyzed glue because the joint has to withstand the stress of pulling the stem out of the pipe. On the one hand, you don't want something that permanent glued into the stummel, on the other, you have a tough physical location to deal with.
You could, of course, resort to hide glue and face this situation again, some time in the future.