I agree that this sounds like a problem that goes far deeper than piano lessons and seems to have a lot to do with self-esteem and confidence. If the older sibling is all things successful and your pupil knows he cannot compete, one way to get attention and have everyone feeling sorry for him is to cry. His school teachers are ignoring him and that is one way to cope, especially in a large class. Someone, somewhere has clearly convinced this lad he's a failure and he's going to live up to it - I would think it's become a habit that may, ultimately, need professional help to cure. It is quite unusual for a child of this age (especially a boy) to cry so readily week in, week out.
However, you are wanting to work 1:1 and so the problem is, in some ways, greater for you. Have you tried a sticker chart reward system (or something similar)? I have done this with "problem" pupils in the past. Set small achievements to be mastered and, when they are, reward with a sticker on the chart. Get a row on the chart completed and the reward is perhaps a little more tangible (you would know what would be appropriate and acceptable). I keep some of those cards boys like to swap in the playground to hand and some of them work their socks off to get one of those. Some respond to the occasional lollipop etc (but that depends on your view and the parents' view on sweets), some to a special certificate..... all sorts of things but something that is for him and
not his brother.
A sticker if he manages not to cry on arrival at lesson and so on. Gradually, of course, the tasks get harder and longer to achieve but you have to build to that. I wouldn't do things "over and over" right now. None of us like that being done to us in certain circumstances and, even at his age, that kind of pressure may be too much for him. Yes, you want him to play the piece correctly (or whatever he's doing) and so, I hope, does he but I would personally use short bursts on things and keep the lesson moving on pretty quickly with plenty of musical games too.
I suppose, if he really cannot stop the tears then he may have to stop lessons but that would be rubbing salt in an already sore wound that he is "no good". I think you have a unique chance to help this chap and I do hope it works out OK for you both. Best of luck.