How to replace motherboard

Replacing the motherboard in your phone is fairly straight-forward, at least as far as phone repairs go. Although iFixit doesn't have a guide specifically for that repair, you can use the Screen and Battery Assembly Replacement guide to get to it and remove it. Obviously there'll be steps you'll want to skip, such as removing the daughterboard, but other than that, the guide will get you there; just use some common sense with regard to which parts need to come out.

Although I'm inclined to think that two separate shops coming to the same conclusion that the motherboard is at fault is probably right, personally I would be tempted to try replacing the battery and see if that fixes the problem. Without those diagnoses you got, that would be the first thing I would have tried. Batteries aren't terribly expensive; certainly they're far cheaper than a new motherboard, so I'd give it a shot, but of course that's your decision. If you want to try, here's the iFixit guide for that repair.

Samsung Galaxy S24 Ultra Battery Replacement - iFixit Repair Guide

I am a bit confused by your warranty situation; you said you haven't had it for a year, which is the normal length of the warranty, but you started out by saying it's a month out of warranty, so am I missing something?

Anyway, good luck with your phone, and be sure to come back and let us know how it all turns out; your experience could help the next person who comes along with the same problem.