Hmm. Seems to me that the simplest thing is to just add a sentence to say whether or not the ability to activate while exhausted also overrides the general prohibition against taking a free Move action while exhausted (if we assume that is how the normal rules will be settled). So:Berserk - This model may activate while in the Fatigued or Exhausted states, but may not take a free Move action while Exhausted. In addition, once per activation, this model may pay 1 Favor and suffer 3 damage instead of gaining a fatigue.
I don't see any other way to specify that the ability to activate while exhausted doesn't grant the ability to take the free Move action as if it wasn't exhausted. It will always be ambiguous if you don't specify it.
Unless the main rules end up saying that you can do the free Move action while you are in the exhausted state, you just can't start an activation while exhausted (unless you have Berserk). Then a Berserk model can activate and get a free move while exhausted all day long.