NAIA ManualIf he stays at a NAIA school, the rule is he must not have competed in more than 8 games (for baseball). Article V section M1. So he would be eligible with respect to the number of games. The physician does need to say the injury is "incapacitating". The request for the hardship waiver must come from the school, and you'll want to get the coach's committment to do that.
For NCAA, the allowable number of games varies according to division, but 5 is below the limit for all 3. Furthermore, the NCAA requires (unlike NAIA) that the games all be in the first half of the season, but that isn't an issue here either, so I think he also may request a hardship waiver at NCAA schools. NCAA does require that the injury be incapacitating, and also season ending. For JC transfers, the NCAA requires that the NJCAA or other sponsoring body themselves have issued a hardship waiver. They don't seem to require that for NAIA transfers, but I think it would still be desireable to get the NAIA school to request a waiver, unless he transfers following this season.
Pay attention to the wording on the physician's report. "Shutting down" isn't forceful enough. And if the player also transfers to a different school, it invites the inference that he wasn't going to play much anyway, and that weakens the hardship request.