As ever, a reminder that I'm not a rules team member and questions can be addressed to them directly with a rules question thread if necessary.
In the spirit of helping out without overstepping my boundaries, here's what the rulebook says - winning a shootout is about having the remaining posse at step 7 (after step 6, Run or Gun), this is distinct from winning an individual round. To my reading, there can't be a shootout winner until Step 7, as that is the step that defines the winner. Sorry if I've misunderstood the question.
"Step 7. Chamber Another Round
Both players discard their draw hands. If only one posse remains, that player wins the shootout. If both posses have been emptied of dudes, neither player wins the shootout. If both posses still have at least one dude each, go back to Step 1 to start the next shootout round."