DS lied about his age to get his gym membership
OK you need to jump on that. And I'm surprised the gym hasn't already, given that you've spoken to a PT there about him.
If he lied about his age, and the gym has an age cut-off for membership, their safety responsibilities and public liability insurance likely won't cover him. So if he hurts himself (or someone else) the gym would not be liable or responsible in a way it would if he was a legit member.
I'm a personal trainer (some of the time), and I work in gyms. We take member safety extremely seriously and I would not want to be in this position. You need him in an environment where he is a) allowed to be and b) sufficiently protected.
I wonder if you need to go about this a different way. Find a gym that accepts junior members; maybe a council-run leisure centre or similar. Get him signed up to that, and find a trainer who is qualified and cleared to coach under-16s.
There's nothing wrong with him enjoying weight-training, and there's nothing wrong with him deadlifting 85kg (provided he's doing it properly).
There is something wrong with him lying about his age to get a gym membership. He is not in an environment where he can be appropriately monitored, protected and safeguarded. Him being there under false pretences is wilfully misleading the people who work hard to make the gym a safe and appropriate place for adults to work out.
Please, I would urge you to step in here.