No he wouldn't.
He's an opportunistic weasel and he knew that the Starks were bound to be defeated. Lannisters had secured an alliance with Dorne and Reach, defeated the Baratheons. The prospects of a Stark victory looked even more bleak when Lord Glover's Army got destroyed in Duskendale.
The slight on his house's honor by Robb's betrayal sure played a role but he has suffered countless such slights before. So why was this one any different? Because he knew that Robb was doomed to be defeated and he did not plan to be on the losing side.
No House is inherently evil or inherently good. Nor is bravery hereditary. Freys fought bravely alongside Starks by accounts of Edmure. But why should they keep doing so? No reason at all.
Walder Frey did what he thought was best for house given the circumstances. Fight on, be defeated, be forced to pay hostages and reparations or possibly lose your head and lands in the bargain. Or you could switch sides and create a larger fortune for yourself and your house. Which one would you choose?
After all, Tullys got Paramount Lordship of Riverlands by betraying their liege lord King Harren the Black and switching side to Aegon the Conqueror when they saw that there was no victory against dragons. It's not the first time they did it either, Before Harren, Tullys betrayed King Teague as well by inviting Storm King to conquer Riverlands. Why is it wrong for Walder Frey to do exactly what Tullys themselves did 300 years ago?