I think that I would leave the decision up to your DS5. Local tournaments are no big deal at all and they are commonly rated by the US Chess Federation. There are some children 5 or younger competing. The tournament is separated into categories based on rating and grade levels. Obviously, you would not register your DS5 to compete against high schoolers at the highest ratings so he should be competing against elementary aged beginners. He should also have enough common sense to walk to the correct (labeled clearly) seat since parents are not supposed to be in the room during the games.
I would not freak about the club coach, who is probably trying to be helpful. Our school doesn't have a chess club, but in the beginning chess classes that DS/DD took, the coach told all the kids near the end of the session to try a tournament if their parents will take them.
As far as private lessons, I have been told that while a child may do well enough, he is unlikely to win tournaments unless he has coaching. I can really see why that may be the case. DS is really unpredictable. He has beaten a high school player who placed in at least one state tournament but lost to a beginning player a year or two younger. He has flashes of brillance but haven't bothered to learn all the basic strategic plays. DS refused priviate lessons because he did not want to practice but only play as he pleased.
