I think usually the trouble you get into is with the term "lesbian" which is seen as a term reserved only for women attracted exclusively to women (not capable of attraction to men).
The typical way I've seen getting around this is a broader term being used instead like "sapphic", so calling it a "bi-sapphic" flag might make some people hung up on "lesbian" happier.
But even "sapphic" gets complained about because then people who identify as lesbians and not as sapphic can feel like they're seen as not using inclusive language, the way sometimes people have been criticized for calling themselves "bisexual" because it can be seen as transphobic (the implication is that you should identify as pansexual and not bisexual, because you should be trans-accepting in your attraction; but other people just reject that "bisexual" was ever really transphobic, etc.).
Controversies aside, I'm definitely more than incidentally attracted to men (probably like 60% attracted to women, 40% attracted to men at this point), but I really relate to your experience - in practice I'm just a lesbian, I'm only going to date women, I'm not going to date men, and my attraction / sexual orientation feels irrelevant and separate from how I actually live and present, which is as a woman who is in a relationship with a woman.
Other people see me as a lesbian because they know I'm married to a woman (and ngl, a big part of this is that I pass and don't disclose my trans status), and I'm pretty much happy with that designation. I've felt like a lesbian most of my life, as weird as that is to say, and even if I'm "technically" sapphic, I live my life as someone who has only ever dated or been with women and who prefers to keep it that way.
So yeah, I get it - makes sense to me!
I probably won't use the label or flag (esp. since it will be confusing and require explanation).
I'll probably just call myself a lesbian because it's what most people will understand.