Which brings me to my conundrum. Is it wrong to identify with a label if there is no way of achieving it? Like if there is no surgical possibility of transitioning to the body that I want, am I still trans?
It's absolutely not wrong. Labels aren't about what you do with your body, it's about how you feel.
As a nonbinary person, I officially give you permission to identify as nonbinary, trans, or both. 🤝 I like 'nonbinary' because it's really broad and inclusive, and doesn't have some of the connotations that the term 'trans' has (even though nb is under the trans umbrella) I think it would be very reasonable if you decide that it fits you.