The commander is under no obligation to allow you to go to the gym. The commander can direct you to conduct PT at the unit location within the limits of your profile. They may even assign you a supervisor responsible for developing a routine for you during this time. The only limit is that you cannot be directed to do something in violation of your profile. The command could improvise and put together a series of sandbags of varying weights is they desire and allow you to lift those or they could eliminate the requirement to lift and require you to do a series of exercises as outlined in FM 7-22 that do not violate your profile.
You might consider a discussion with the CDR/1SG on the merits of allowing you to work out in the gym and how this workout is in your best interest and in the best interest of the unit. Invite them to drop in on you and check you at any given time. Consider bringing them a workout program and have a way of recording it, you could offer to sign in and out of the gym. Consider going to the gym at a designated time and return to the unit at a designated time to show the command team you are serious about the exercise session and not trying to get over.
If you are sincere the command team may give you a shot at you managing your own program. Keep in mind most chains of command usually do not support these programs because Soldiers have failed to follow through on their responsibilities and the chain of command does not want to get burned. Do your homework come up with a good solid plan that allows for spot checking, can be verified, and one in which can show results in terms of building yourself physically and improving your APFT score.
Just my thoughts on the matter. I hope this helps! At least it is worth a shot, the worst they can say is NO! Hopefully I have armed you with information that makes it hard for them to say No because you have addressed all of their concerns in your plan.
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