After coming as far as you have in the tournament, it would suck big-time to be disqualified because of an uncertainty about the rules. Luckily, there's a table of official-looking types set up to one side of the waiting area, who you could ask for clarification - and so, with Briar, Cordelia, and Ayane following along, you wander over and do just that. You pose the question without mentioning your fairy companion, as you've yet to see any evidence that the men running this show are in any way aware of her existence.
You're told that, for reasons of legal accountability, the tournament prefers that all medical treatments be carried out by its own vetted and accredited staff, or by pre-registered personal aides. Using inherent abilities or learned skills to heal your own injuries is fine - there's a disclaimer in the entry forms that absolves the tournament of responsibility in such cases - but they would much rather not have the contestants going about attempting to fix each other's injuries. The odds of doping, deliberate sabotage, and/or accidental injury are simply too great to be ignored, especially in the children's divisions. As it happens, you're free to request a medical evaluation at any time other than in the middle of one of your matches. You can even visit the infirmary after every fight, if you want - some of the healers would greatly prefer it if everyone did just that, since it would let them keep an eye on their patients and ensure they weren't going off and doing more stupid, self-destructive things in the meantime. If the experts deem that medical treatment is warranted, they will provide it. Similarly, if your personal doctor, faith healer, shaman, guardian spirit, or what-have-you decides that a certain method of care that has been pre-approved by the tournament committee is needful, they can provide it. Unregistered healing, however, is a no-no. Again, legal issues.
This was all outlined in the registration forms - which Lu-sensei handled for you, and never got around to explaining - and again in the rules pamphlets - which you couldn't read well enough to make sense of Japanese lawyer-talk. The officials you're talking to appear startled when they discover your ignorance on the subject, and make a point of reading the relevant sections of the rules back to you, translated into everyday terms that you, Cordelia, and the unseen but intently-listening Briar can all make sense of. Ayane nods along, confirming their words.
Gained Japanese E
To sum up, it seems that having Briar heal you would be allowed, except that as far as you know, she isn't registered. Likewise, the little touch-ups of healing magic she's administered to your friends after their matches were technically illegal; it's just that, since everybody walked out of the ring under their own power, nobody was concerned enough to look closely and catch what was going on.
"Do you wish to request a medical evaluation, young man?" one of the officials says, as the final match of the round comes to an end, with Gorn finally managing to pitch Morgiana out of the ring.