Okay, so, within current canon, I do not believe for a second that Katrina would enter into any kind of romantic or sexual relationship with Ash. He was held captive by the Klingons for seventh months, tortured, and sexually assaulted. He needs counseling.
[Aside: everyone on Discovery needs counseling and I would subscribe to a whole second streaming service that was JUST various Starfleet officers in therapy. Something I really love about Disco is it’s focus on trauma and consequences. I want to see the recovery process, too. For all the series. Individual therapy. Support groups. Give it to me.]
She would also caution Gabriel against it. Not only are they both traumatized, Lorca rescued Tyler, and is now his commanding officer. It’s not against the rules, but Ash needs time and space to heal, not complicated relationships. “I’m your friend and I’m not going to tell you what to do, but this is a bad idea.” [Insert Lorca’s #unimpressed face]
Now, Ash is integrating into the crew pretty well, so upon observation she would say, “But your instincts were good pairing him with Burnham.” They are also both traumatized, but they are more like equals (they don’t know he pulled rank on their second date, but also I don’t really know what was going on in that scene/on that planet so I’m reserving judgement), and Lorca (and in this scenario, Cornwell) can keep an eye on it. Plus Tilly is GREAT for both of them.
At a time where everyone is healthy and stable, or at least more so, maybe? Tbh, I’m having trouble seeing Lorca and Ash in a healthy relationship. It would take a lot of time. And honestly, Ash deserves better. Kat deserves better, too, of course (does anyone deserve Lorca the human disaster? No, but lbr that’s half of why I love him so much), but she set better aside years ago. In this sense Kat & Ash teaming up to love their human disaster boyfriend and realizing THEY can have a nice, friendly, healthy, stable, cuddly, let’s go fishing type of relationship is a cute idea. A “let’s share this burden because this is exhausting but he is actually worth it” type deal leading to “he forgot your birthday but I didn’t” and then Lorca’s like “you guys should fuck, can I watch?” and much eye rolling but then then it’s in their mind and an innocuous brush up against each other becomes charged and suddenly they are escaping to quarters and Lorca’s all “I told you so, now make room for me.”