All are possible!
It wasn't that my mother and I didn't care about Spike; it's just that he broke things faster than we could fix them -purely past event. He broke things faster...once...but now he may not.
It wasn't that my mother and I didn't care about Spike; it's just that he breaks things faster than we could fix them -he has a habit of breaking things faster than you could fix it. The habit was there in the past as well!
It wasn't that my mother and I didn't care about Spike; it's just that he would break things faster than we could fix them -'would' also talks about his past habits. Such constructions are common in storytelling (say I'm describing my love... Mary and I would sit on the bench and enjoy the lovely sunset)
But you said that the event belongs to yesteryear. So...it's in the past. Then let's make everything in past!
It wasn't that my mother and I didn't care about Spike; it was just that he broke things faster than we could fix them.