By yours truly and @hyperfixationwizard (here to attend to all your trans Soap needs)
Think about how there's a chance Soap hadn't settled on a name yet before he met Price.
He joined the army pretty young—hell, this makes the part where he lied in his forms make even more sense—and every time he'd just try a different name, nothing really stuck out to him yet. So when he first enlists he just goes by MacTavish anyway and figures he'll make something up on the spot if anyone ever asks and tries not to think about it too much.
It doesn't take long of course for "Soap" to become a thing and whilst it's a stupid name, secretly he's a little relieved. And then he meets Price and his world is kind of flipped upside down because Price is kind and supportive and will listen. And Price does everything in his power to help him and protect him and well... It doesn't seem too difficult to settle on what name he wants anymore.
So he makes up his mind and gathers his courage and finally goes to tell him that he wants to change his files. Price of course asks to what and it takes him a moment to find the words but at the end he just blurts out "John, sir." Price doesn't give a sign of acknowledgement and neither of them comment on it at that moment.
Price hasn't said it to anyone but it did stick out to him as a possibility and he'd die before he admitted it but he did cry that night.
