Is it possible to tell if a doe has another kid in there with doing anything too invasive?
Yes, once your doe has kidded, the question always is, does she have another kid in there? It's a good thing to know, and believe it or not, there is an easy way you can check. it's what I call "bouncing".
I have heard this called "bumping" as well, but I don't really care to call it that because it gives the impression of doing this procedure in a ruff manner. It is done smoothly and gently.
This little trick only works after the doe has had a least one kid. This is because, at this point, her belly muscles are relaxed, allowing you to actually feel the kid(s) inside her. Wait until the first kid is fairly dried off and then try this technique.
How to "bounce":
The doe needs to be standing to use this technique. Stand behind the doe (facing her butt). This can be a little messy, but at this point you're probably covered with birth glop anyway, so don't worry about it. Bend down and reach around her belly, like you're gonna' give her a big hug, and place you hands, palms against the doe's belly, right in front of her udder. Now pull up gently and "bounce" her belly. If there is a kid still in there, you will feel some hard, knobby, solid stuff (a goatbaby). If there are no more kids left, you will feel just soft, squishy, goat guts. Move you hands closer to her rib cage and try it again. Try it a few times to make sure what you are feeling.
Practice is key here. Try this on all your does as they kid and once you've felt what it feels like "with" a kid and "without", you will always know.
You will quickly become an expert at bouncing. I have had 100% success with this technique. It sure is nice to know the doe is done kidding. Now you can both relax and enjoy the goatbabies that were just born.