Before I start answering, I must include the reason for such a name the character bears, which I unintentionally omitted from the original comment.

"Agatha" is her real name, with which she grew up at the orphanage, but the "Black Sun" part comes from the fact that she was born during an eclipse (doing calculations for it was fun). After learning that, her little brother, having fun with his own heritage (to which he wasn't entirely attuned), gave her an "Indian name". Agatha liked it and said that, if she's ever to lead a tribe, that's what her name would be.

instances where she'd be forced to drink alcohol, or fake it, perhaps

One time, Agatha visits what its inhabitants call the Dome. The Dome is one of smaller hubs, with a couple of dozen survivors as a permanent staff and a few that are passing by and staying only for a while; used to be a typical five-storey house, until cluster rocket blew off its roof and highest floor as collateral damage, and it's also a set-out point for a major plotline.

Agatha manages to get in minutes before one of the hordes - very populous groups of zeds who, due to the way their system works, lost almost all of the sense-making that may have been left had they been alone - rush through the area. The Domers hope to wait it out, until someone makes a noise loud enough to get the horde's attention. Most zeds mindlessly rush forward to whatever destination they may have had, ignoring the Dome, but those closer to the noise got very aggressive, so the Domers had to fight back.

After this particularly terrifying and difficult battle, those Domers who had no responsibilities at the moment - which included the Dome's leader, Ellis Gatling, and her leuitenant and one of the best marksmen in the state, Chris King - decided to celebrate the victory and get some rest, which included drinking some stored alcohol. Agatha had to politely refuse, explaining the situation. Ellis doesn't believe it at first, but Chris tells her how he served in Turkey with an Asian guy who'd be a hell of a mess after what Chris himself could take in and have fun with. After that, the relationships between the leaders of the Dome and Agatha go much smoother, with her being able to open up to them.

or develop a love interest

Indeed. I haven't fleshed that out yet, but there's certainly a plan to portray this part of Agatha's personality.

or times when her crossbow comes in particularly handy (since you describe it as a vertical crossbow, I wonder if there's an instance where it might be necessary that it was such, rather than just being a cool visual)

Well, she is living in a world where any amount of shooting firearms will attract the whole gang of the local zeds (who, I will remind you, communicate through wireless connection and relay such information to those in the vicinity), so using an almost-silent crossbow is a very effective method of concealed killing.

Vertical crossbows are usually lighter than the classical wooden or solid-metal ones, because the vertical constuction is a modern invention to which the industry added a few of helpful sidefeatures. For example, one can disassemble such a crossbow if need be and reassemble it to the precise same use. The lightness translated to easier holding when aiming and, thus, higher accuracy, which is important when you're hunting either animals or living humans (zeds or otherwise).

AND THAT DOGGY BETTER LIVE!!!

I love the doggy too, but given how the setting is Stephen King-inspired, I'd have to see about that.