Moving him to a third school doesn't seem like a good idea to me. Changing schools is disruptive and there is no guarantee that the situation would improve. Thi sounds to me like an attitude issue or, as mummytime suggested, possibly SEN.
Firstly, I'd talk to DS2 to find out what he likes and dislikes about his new school AND, now that he has something to compare it against, what he liked and disliked about his original school. Even though it is obviously a consideration, be really clear that the money aspect is not what is driving your decision, but that you love him and want him to be in a school in which he can be happy and can thrive. If you pitch the conversation well, you should be able to get across to him that you are looking for a change. Whichever school he prefers, talking about this in the context of a possible school change might be the shot across his bows that he needs. It could be that he is miserable because he actually prefers his original school. If so, he may find it easier to cope, and give him an incentive to keep his nose clean where he is whilst he waits.
Secondly, if sport is very important to him, I'd try to take that out of the equation by finding a club (or two!) in which he can excel in a team with other local children. Our son loved rugby and enjoyed swimming, and both of his clubs had an equal mix of members from state and private schools.
Have you burned your bridges with the original school? If you wanted to move DS2 back, presumably DS1/DD1 is still there, so you would have sibling priority on the waiting list. Could you sound out his old head of year about the possibility of a return, and how DS2 could be helped to settle back in, given he would be coming back with a different attitude having seen that the grass wasn't greener? If the disruption had been tied to particular individuals, perhaps he could be put in another form or house?