You're correct in assuming that identical abilities, powers, spells, etc... do NOT stack.
Since the rats are already slow by nature, when summoned, they have one (1) action they can do.
Essentially, you can activate a swarm of rats, do their thing, get them killed/sac 'em and then pop out more rats that can then activate ad nauseum.
It's pretty stupid actually.
I guess the trick is to kill Nix, the catchers and the stolen as quickly as possible and then go for Hamlin. Or do your best to ignore them all and go for your objectives.
Either way, I think Hamlin needs a huge Cuddling. He is not fun to play against.