Guys, stop focusing on a yeshiva.
They had their chance to educate the kid and to show chesed if the child is a difficult one. There are numerous stories that gedolim did not tolerate expulsions, such as R Schach telling the yeshiva managers that it is ok to expel the the kid and then was seen personally tutoring the kid, leading to them taking the kid back. I am not saying that there are no kids that should be elsewhere, but most cases are not.
So, it is either the yeshiva managers (1) are not aware of the position of the gedolim, or (2) are not willing to follow their example, or (3) it is a case of a kid who really does not belong there. My humble opinion is in all of 3 cases, there is no reason to try to get back there. If you think differently, please specify which of the 3 hypotheses above you presume.
So, instead – focus on what the kid needs to develop into an erliche Yid and start providing that. If you think that in a long-term he needs to be in a yeshiva, you can continue searching for the right place, but there is no reason you can’t provide a good environment for the kid yourself with the help from friends, tutors, local rabbis.