SKM, I don't think ambivalence is bad (if it is I'm probably a lot more guilty than you).
1. You say you won't be contacting him anymore and you guys don't talk; 2. You say you want to know when it is time to give up; 3. You say you won't be the one to file; and 4. You say you are ambivalent about divorce.
I think you look like someone who doesn't want to try (which is understandable), or maybe doesn't want to try that same way again, and doesn't want to give up (which is also understandable). What is left other than limbo if neither of you can make a move to the other. I think he made a move. I think it may be your turn. (But of course all that matters at the end of the day is what you think).
I wonder what would happen if you and your H only had a calling or texting or friendly relationship for a while (with you initiating sometimes despite your pronouncement). Seems like he doesn't want to be divorced either. Seems like he's at least had a first awakening and walked quite a way towards you. Seems like maybe he was going to use the remaining dog as a bridge, but then maybe got lonely, or it felt insurmountable, and got the new dog.
I think were it me I would stop making pronouncements to him (I won't contact you again) and just try to see if you guys can be friends again. Is he someone you want to be friends with? What if you had the opportunity to figure that out without the pressure of living together? What if you called him up and asked how it was going with the new dog or asked if he wanted the new dog to meet the old dog? And avoided relationship talks at all cost.
You are at a different stage than a lot of folks here. I wouldn't call that an early or false reconciliation in the traditional sense; I think it was just too early for your anger and his lack of remorse. Too much too soon. What would a 180 for your situation look like? How can you resolve your own ambivalence one way or another?