I think it's general practice for T's to be responsible for the time-keeping piece. Every T I have spoken to has been the one to take responsibility for it, and for any overages if they lost track. I've had several student T's, and even when supervisors interrupted to tell them to keep watch of their time, it was always on them, never the client.
I tend to dissociate in session, or get caught up in the topic. At times I notice we have gone over and appologize to T. She always says it's her job to watch the clock...
I think it would be more-so your T's job to watch the clock if he is going to arbitrarily charge over-time. He should at least be alerting you to the time coming to a close, and offering you the option of continuing & needing to pay, or wrapping it up. He doesn't seem too into taking responsibility for it though, so as others have said, your best bet may be to do your own time-keeping.