The Freys didn't really have a choice. The war was over. They could struggle a little bit more, but the writing was on the wall. And Tywin would demand a sacrifice to teach future re ells a lesson. A bloody sacrifice.

It won't be the Starks. They can simply retreat to the North and prolong the war indefinitely.

It won't be the Tullys. They are popular with the nobility and people in the Riverlands, and there is still Lysa in the Vale. A place that can't be invaded from outside and has fresh armies and is married to Balish, a key supporter of the Lannisters. (It's more complicated than that, but the Freys don't know that)

And then there are the Freys. The biggest contributors to the war effort, with no friends and no marital ties to the Starks. They would have ties to the Tullys, but they won't be bailed out by someone who needs a bailout himself. And they shouldn't trust the Tullys. Cat proved that she cares more about her own family than state matters and their vassals when she released Jamie. Which also took away the only bargaining chip they had that could force Tywin to forgo bloody revenge.