I think the name wouldn't show up on the other person until they embraced their real identity and the new name. It's not like the names have anything to do with your government name. It's more about identity. So I definitely think it'll be the new name that will show up.
At least that's what I'd like to think personally. It would be too sad if people had fated partners with the name they don't identify with, tattooed on their bodies forever.
Or maybe, since the whole name thing revolves around fate, destiny, fortune or the likes, the "namers" would be marked with the current name of the person they're destined to meet. Just like it's their fate to find eternal happiness together, it was written in that person's fate to change their name for whatever reason, including having a new gender identity.
Considering how it was pointed out in an earlier chapter that, in this AU, gender doesn't matter much when it comes to relationships and soulmates (?), even in the eventuality of the namers finding each other before one of them transitions, their destiny would have calculated that outcome too.
But I'd say, we're probably just going on about a loose thread of the plotline that just wasn't considered by the author.
I thought about this too after I posted the previous comment. It's definitely a better explanation but that's only considering names really do mean they're soul mates and this is fate's hand. But we really don't know enough right now because our pretty boy MC doesn't give a shit about all this and his gorgeous boyfriend is so goddamn skeptical and suspicious of it all so we really don't have any answers for now.
And about the gender, so far MC said gender is inconsequential but ML said it's something that's a deal breaker for some people and that's why they hide their status too. And tbh I'm not sure how much we can trust the MC given how he doesn't know he has a name on him. And he was raised by scatter brained parents too. So pretty much everything we know right is not much to go off of.
We just have to wait for the author to put out something more.
Yeah, both the main characters are unreliable narrators, so we're actually discovering the world building along with the plot, rather than being introduced to this universe and its characters from the beginning. And I can definitely appreciate this more "active" way of interacting with the story instead of being spoonfed the lore.
For example, when trying to check if the name of his mark could have been the MC's, the ML asks the MC if he ever changed his name. That would dispute my point. However, can we trust the ML knowing all the rules that apply to the phenomenon with how skeptical he is? And with how the phenomenon has been sensationalised, can we assume that cases like these have been observed and analysed?
We also have to take in consideration that this is a translation. Some information that the translators might have considered minor, but was actually more useful details, could have been lost in translation along the way (or at least that's my case since I only know a couple of Korean words and phrases).
I'm still not excluding the possibility of the author just not considering these particular cases in the storyline. Trans people are very rarely mentioned in manhwas, at best we see "crossdressers", and even then, their expression is more of a plotline rather than their real gender expression.

i wonder how this name thing works for trans people with deadnames