Yes I think given the importance to you of getting it working again, I suggest setting aside any attempt to get the builders to do it or even allowing them to be involved in any way. Their primary focus will be on how to swerve or minimise the cost to them, not how to make you a delighted customer so any input from them will be diversionary and will just muddy the waters.
So I suggest marshall your resources and 1) get the problem fixed then 2) once fixed turn your attention to getting the builders to pay for it, if you have the time and the energy. Otherwise chalk it up to experience and keep a close eye on your alternator temperatures in future!