My first uneducated guess is that it would be much harder to create collisions, but not impossible.
My second uneducated guess is that it would be impossible to create the massive kind of multi-collisions required for a HashDoS. That is, an attacker could possibly create two or three messages that hash to the same value regardless of the seed, but not hundreds or thousands. And since a moderate amount of collisions are expected anyway in any hash table and cared for, the attack would fail.
Am I correct about that? If not, would it help to mix the seed repeatedly into the state, like here?