Neither sides is inlined to protract the dispute much longer/any longer/any more
As far as the phrases "any more" and "any longer" are concerned, they are interchangeable, without any difference in meaning.
As for the phrases "much longer" and "any longer/any more", though they can be used in the sentence, they are not interchangeable because they convey different senses.
If you wan to emphasize that neither sides is inclined to protract the dispute even to a small degree/in the least, you can use any more/any longer. On the other hand, the use of "much longer" in the sentence conveys the sense that either of the sides can stand the protraction of the dispute by a small amount/degree, but not to a greater amount/degree.