Thank you for the considered reply!
It’s not so much that I can’t see the fretboard, it’s just wrist strain. Tilting the guitar down even a bit makes it quite a lot worse, so I may be doing something wrong. The further down it goes, the more I need to push my wrist forward and arch my hand away from the neck so a) I can play with my fingertips and b) I’m not muting the other strings with my palm.
Getting used to the position and stretching certainly make sense to me, but I can’t see my wrist ever wanting to bend that much…it just doesn’t feel possible for me to play in that position.