Great question!
Let’s exam a previous ruling and follow up:
Joan text:
"Joan has -1 bullets for each other dude in her posse.
Joan cannot flee or be sent home from a shootout.
Joan’s attached cards cannot affect opposing dudes unless they have bounty higher than their influence."
Harrowed
When a dude with the Harrowed keyword is involved in a shootout, that dude can absorb more casualties than a normal dude. You can send a Harrowed dude home booted to cover one of the casualties you take in a shootout. Discarding a Harrowed dude covers two casualties, and acing a Harrowed dude covers three casualties. If a Harrowed dude in play gets aced by a card effect (but not through being aced to cover casualties in a shootout), discard that dude instead of acing them.
Take Yer Lumps:
Each player must try to “fairly cover” casualties as exactly as possible, e.g. you must discard one dude if you have only one casualty (i.e. you cannot ace a dude for one casualty), and you must ace all your dudes if your casualty number is twice or more the number of dudes you have in the posse.
The Take Yer Lumps rule must be followed when dealing with the Harrowed trait.
As Joan can not be sent home, if she is the only Dude in the posse, she will remain in the shootout. This is similar to Mario Crane.
EDIT: Ruling changed to reflect this article:
http://pineboxentertainment.com/2020/01/24/hells-comin-with-me-texas-rangers-join-the-fight-against-the-darkness/