You seem to be conflating a few things:
IANAL, but my understanding of the national park regulations are that only boats with park permits are allowed to use the park for commercial purposes...and yes, if you're paying someone to use their boat, that's a commercial charter -- the word "private" in "private charter vessel" simply means that the paying customer isn't sharing the boat with strangers.
Separately, the CONAP regulations seem to state that all visitors must pay the fee, regardless of the ownership of the vessel (subject to exceptions for senior citizens, the disabled, etc).
https://descubreanp.conanp.gob.mx/swb/conanp/ANP?suri=4
Did you ask?
When I've been on Aldora boats, there's always been an analyzer...though few divers ever ask to check their tanks, so no one is waving it around, but it's there.
It's been a while since I dove with 3P, and my vague recollection is that they had an analyzer in the shop for customer use, though they may also have one on the boat.