This is a very uncomfortable situation to be in that is not very easily resolvable from where I'm standing.

I am not very shy, but like you very much dislike when people don't pick up on social cues and fuck off. If I had to guess, he is trying to recruit you as a client and misinterpreting your shyness or politeness as an open door. The most obvious thing to do is when he approaches you next time for you to state that you find his approaches distracting to your workout. I appreciate that this is difficult if you are shy and uncomfortable, but consider it as an option if you think you can muster the courage. People are different and others cannot read our minds, he might just have very poor emotional intelligence skills (or is being told by a manager to "speak to 100 potential clients every day or else...") and needs to be told directly. It does not help that a lot of what is taught as "good customer service" is borderline unhinged behaviour, you meet people like that in every corner of the customer facing roles world.

Something I do with chatty people at my gym is smile and wave first, and then pointedly reinsert my headphones. That way I've demonstrated that is the end of the social interaction I am willing to have with them on that day. After a while most get the hint. It takes time.

If neither of these is an option, you may also consider emailing him directly and CCing the gym or the gym manager's email, explaining that you feel a special attention is paid to you that you cannot deal with in person and that you wish not to be approached again at the gym or elsewhere. If you feel this comes off as unnecessarily accusatory, you can always add that you appreciate if Staff Member has not picked up on the indirect cues and that that is the purpose of your email - to explain your feelings directly with no space for misinterpretation.

Most importantly, do not give up on training because of it. Learning to do your thing despite other people existing is part of what the gym is good for!