Gadget Dudes + Fort 51

If I invent a Gadget Dude as part of set-up, can I use Fort 51’s React

React: After you invent a Gadget Weapon or Gadget dude, give 1 bounty to an opposing dude with influence less than the Gadget’s printed cost.

Is it then used during the first turn, so I can’t use it again?

Yes and yes! See ruling on Post-A-Tron and Gadgetorium.

However, in the current Doomtown card pool, there are no Gadget Dudes that Fort 51 can start in play.