The debate I'm having with pets is that, for the most part, there the rarest thing a boss will drop (disregarding jars of decay, sand, etc.) I'm still training my stats for bossing and I've been thinking about what would be best to do with pets. The chance of me getting them during the initial grind of going for items are technically better than getting the jars. I'm leaning more towards staying until the boss is fully complete via the completionist log.
I think that you should do completion logs for one boss at a time, including pets. That, or you can always do pets as another series after all non-pet drops have been earned. Best of luck!