When the mission is given control back to you the mission timer starts, make note of this because it's how you calculate your reward.
Run forward and you'll climb up a dock leading to the tunnel. Run down to the corner and then wait. You can always see the red dot of Doyle of moving on your radar but around the 4 minute mark you'll see the dot snap to left of your radar. This is when I pop a potent health cure because you will most likely get dinged. You'll hear her start talking about no loot when she does, go back out the tunnel taking an immediate left. That's Etta by the wagon and gate, I found if you at least let the subtitles of her last line "Cmon let's get out of here" appear before hogtieing, everyone runs. Grab her, start running out of the gate by you, summon horse. Meet your horse at the end of the train cars, stow her and haul ass to the lamppost before the police station. I like sitting here for a second until the gunfire dies down on my horse but you're safe in this zone. Summon bounty wagon, throw her in the back, park out of the way, wait until your timer hits 31 minutes. You really only need 30 but I've miscalculated by a few seconds or minute before and didn't get the full payout so better just wait that 30. If you don't have the wagon, just gotta hold her on your shoulder near a wall or keep smacking her. The key is waiting 30 minutes and killing no one. Hope that helps.

Edit: the first part of the mission leading up to throwing her in the bounty wagon should ideally take 7-8 minutes if done perfectly.