First off al, it's really very generous what you are doing here. It's inconvenient for you, and even if it wasn't it is still such a huge help for your friend and surely not everyone would act like this.
And here's the deal; he does not realize it.
If he was grateful/thankful for this and would do his best to achieve his goal as soon as possible, everything would be alright. But with his current behaviour, it's just exploiting you.
If he lived in his own house/apartment, then it would be his own decision, yes.
But after he has much fewer costs while staying with you, I think it's not acceptable and it's sad that it's even coming up.
I would definitely bring it up in a conversation. I don't know what a type of person he is and how he'll react (if he's a defensive type who'll shut down, it won't be much effective I assume).
Still I would point out to him that you really are doing him a big favour, that it's inconvenient and risky for you, that you love helping him out but that his actions are not correct.
(and by the way, he's not even helping with cleaning, cooking or something like that? Come on!)
Depending on the type of person he is, you should consider his way to react in discussions and either confront him directly, or bring it up slowly. If you make him feel guilty it could either give you the results you want, or he gets angry and shuts down.
But he either has to leave or pay more rent. And fuck it if he has money to travel around, he has money to pay the normal amount for a rent. That's it!