The run would have to take place over two saves since some dogs are inside the giant bathroom and some are in the lost village/cerebral falls.

The blink could still be included. Its chance isn't entirely random, it's about 60 times more frequent in the CORE (the white river) as discovered by knilnoot:

https://steamcommunity.com/app/2134320/discussions/0/603026374004851632/

And demonstrated by radioJim: