Just FYI - I'm pretty sure that "courting gifts" have nothing to do with romancing and are just another type of gift like "trophy" or "imperial memorabilia"
I've heard this before, but that's not always true. It seems that some characters only accept courting gifts if you are romancing, some don't care, some only accept them from the opposite sex, some don't care. It's kind of random.
I have a male Bounty Hunter and I decided not to have the romance with Mako, and now she doesn't accept courting gifts. On the other hand, I have a female consular and I declined to romance Theran, but he still does accept courting gifts, and I think he also accepts them from male characters.
In case you're interested, according to info I got online here's a list of companions that accept courting gifts from both male and female characters. It's quite random: