Who is legally bound to pay the fine resulting from the old car's installments, and is it permissible to share it?

A late fee is not permissible in Islamic law because it is a form of riba (usury) from the pre-Islamic era. Therefore, a void condition or stipulation does not have to be fulfilled and is not permissible. A person must strive to avoid it by all possible means. If compelled to pay it, there is no sin upon him, and the one responsible for it is whoever caused the delay.