I feel like there might be something else going on here, does he need extra support for a reason? I'm just thinking back, and I was a summer baby so only just 18 and basically got on with uni life, and was fully responsible for all travel, accommodation, life costs, applying for a loan, cooking, washing etc etc,i called my parents once a week max and asked nothing from them.
It sounds like your son just needs some additional help, especially socialising, in which case I think you're being a lovely mum doing this, and its the right thing to help him settle and build some good routines, and make some friends. I think it's just about discussing with him, how he pays for this because you cant fund it right now, but also making him aware that the average student at the club will not have their mum do lifts for them - and he might want to think about that and how he manages it. If he is not comfortable paying, then he is making the choice to sort himself out, or not go. That's on him, not you. He also needs to think socially about how it looms potentially, and work out if actually working out the transport to this club is a great chance at trying some independence.
If I've got it wrong and there's no reason he requires any additional support, then I don't really get why he's signed up for something without thinking through transport and as a young adult with a job and a bursary, he needs to cover the cost and realistically surely he won't want to rely on his mum?