What you are looking for is an orthopedic process which can polymerize a structure according to the properties of the tooth enamel without reacting to the properties of the gum or other soft tissues near the tooth. The end result is intended to be a protective ductile covering over the teeth. This needs to be done in no fewer than two steps.
Differentiate between tooth enamel and other mouth parts
The retainer must first find some way to "know" what is part of the tooth, and what is not. Typical retainers today rely on the orthopedic practitioner to visually look at where he or she is applying the compound. They can see the teeth, and use their fingers. That is a very complex apparatus, to be sure, and they do cost quite a bit to hire. So how can an amorphous compound "see" what is a tooth, and what is not, to save us the expense of the orthopedic surgeon? There are generally two options available when you can't use eyes for this distinction: Chemistry, and mechanical properties.
- Chemistry: Tooth enamel is the only part of the mouth that is comprised of calcium phosphate, Ca$_3$(PO$_4$)$_2$. The first thing your blob would need to do is identify when it is touching this enamel, by a chemical reaction. But you want to be able to remove the retainer, so it can not form a bond with the enamel. The only way to do this chemically is with a two part epoxy: the resin layer identifies the tooth enamel, and the hardener layer forms the retainer. For the resin, this means you want a chemical agent that is catalyzed by the calcium phosphate. Essentially, what the athlete would be doing here, is to start with some sort of chemical paint that they rinse in their mouth, and this paint polymerizes when it comes in contact with Ca$_3$(PO$_4$)$_2$.
OK, I do not claim to be an oral health research lab, so this won't solve your problem in real world with any specific chemistry. Let's just leave it at your inventor has discovered a nucleophilic phosphine catalysis paint which polymerizes when it contacts the phosphate in tooth enamel. That's what those words mean. Nucleophylic means "loves the nucleus of..", "phosphine", and "catalyzes." Will this taste good? I think it will not, no. Check mark for disgusting. Will it be safe? Imagine your inventor figured that out, because (s)he has the research lab. Stack Exchange has been unable to fully staff their research lab, it appears. On application of this epoxy resin, and after it has polymerized into a sheath around your calcium phosphate, you move on to step two.
The second phase of the chemical solution would be the actual "gum" which will become the retainer. This gum is a different material from the paint, but is activated by the paint in the way that an epoxy hardener is activated by the resin. At this stage, your inventor is quite a genius indeed. When the resin and hardener react, there will be molecular bonds being formed that release energy. Just like cement and household epoxy, making molecular bonds requires either adding or releasing heat. So you could have retainers which painfully freeze your teeth while they harden, or become uncomfortably warm when they harden. In real life this also happens, which is the biggest reason they take a mold and make the retainer outside of your delicate mouth. However, this inventor has reduced the problem to one of mere discomfort. OK, we are still in the disgusting category. Check.
The third part of the chemical retainer solution is removing the retainer when done. Obviously, our athletes will need to tickle their taste buds once more with something from a lab beaker. This chemistry cocktail converts the catalyzed resin into a new compound which does not bind to your calcium phosphate, with its own new and remarkable flavor. But again, this inventor has solved the problems of safety, so while it may not win prizes at the county fair home-cooked delicacies contest, it will only cause uncomfortable chemical heat or cold, and earn you lots of ice cream credits to rinse out your mouth.
After an amount of time that you decide, the athlete can then pull their appliance out of their mouth and test its resiliency against a power-dropped cast iron kettlebell.
- Mechanical polymerization: Tooth enamel is more than simple calcium phosphate, actually. It gets its famous hardness and durability from the crystal structure that your bone cells organize the molecules in. The enamel structure is a decalcified hydroxyapatite, written as Ca$_{10}$(PO$_4$)$_6$(OH)$_2$. It is layered down by cells mechanically in a way similar to the way snails and other mollusks grow their shells. The solution we will talk about here is in fact a paradox to modern chemistry, because in polymer chemistry, mechanical energy degrades polymeric chains. In nature however, mechanical energy is often used to create new polymers. So why buck the system? If we want to form polymers with mechanical energy, we need something that's alive in your mouth. How a bout a nice, warm cup of bacteria?
Your inventor has grown a bug (I call any small living thing a bug, deal with it), that happily assembles a polymer mass when stimulated by vibrations at a certain frequency and amplitude. Basically, when these bugs are swimming in their happy juice, and you shake them at just the right speed, they make a plastic-like mass. Here, you will take advantage of the tooth's hardness to reflect acoustic energy, and the bugs will do their polymer building. The way the athlete does this is by putting the soft dough into her mouth and shape it around their teeth with their tongue and fingers. They don't need to be exact, but try for full coverage. Once this is on, they need to suck the bugs into their mouth with a straw (you won't be able to separate your teeth at this point, the gum is nicely shaped around them. Try to swish the bugs around the dough, so it can seep in between the tooth and the dough. Are we comfortable yet? No? Good.
Now comes the mechanical energy. This is applied with an ultrasonic transducer. It simply makes a very rapid vibration at a stainless steel disk. The apparatus could be described approximately as an electric toothbrush with a small dental mirror in place of the brush. This is what it would essentially look like. Your athlete needs to rub this disk over the dough, as if they are spreading cake frosting over their teeth. This sends vibrations through the dough, but when the vibrations hit your hard tooth, it echoes off and rebounds. This tickles the bugs near your tooth, and makes them start building their polymer from the dough. The bugs on the dough near the steel disk will also be building retainer material, so you will need to wipe the disk off once in a while. Every 20 seconds or so, you need to check to see if your retainer is built up enough. Poke the dough with a toothpick, and see how deep it goes. You should not be able to touch your teeth. Keep doing this until the teeth have a layer of plastic-like polymer that is thick enough for your event.
There should be parts of the amorphous blob that the bugs did not harden, and your athlete needs to scrape this off. They can now remove the retainer blob, thankfully, and begin brushing away the soft bits.
Now you advise your athletes not to swallow the bugs, and offer them a cup to spit in. Wash everything in alcohol, especially your tongue, to kill off what's left.
In theory this additive process will form a retainer that can be removed and reused, because it is not bound to your teeth like the chemical process.
I do not feel your inventor will be enjoying the high life on either of these patents, but your athletes will suffer in the way your story intends.
