If the need for a single word outweighs the need for that word to have some currency then prick could possibly fit the bill. Prick is more specifically to do with writing down music but has been used just to mean 'to record in writing'.
13 a. trans. To write or set down (music) by means of ‘pricks’ or notes (arch.); also, to write music in (a book) (obs.). Also absol. or intr.
1765 WESLEY Wks. (1872) XIV. 330 They [tunes] are pricked true, exactly as I desire all our congregations may sing them.
The Wesley citation suggests that 'from memory' songs are not as accurately sung as 'prick-songs' are, which to me suggests that it is words as well as notes that are 'pricked'.
The OED also mentions the usage
†14. To write down; to note or jot down; to record in writing. Obs.
c 1400 Destr. Troy 418 Als put is in poisé and prikkit be Ouyd.
You could save prick from obsolescence by resurrecting the usage from Collone's Destruction Of Troy. A slightly longer quote, showing that the use of 'pricked' was not for song, but for fables and stories.
All thies Japes ho enioynit as Gentils beleued,All thies maistres & mo she made in hir tyme,Als put is in poisé and prikkit be Ouyd,Þat feynit in his fablis & other fele stories
For prick-song the OED has
1. orig. pricked song: Music sung from notes written or ‘pricked’, as distinguished from that sung from memory or by ear; written vocal music.
2. esp. A written descant or accompanying melody to a ‘plain-song’ or simple theme; hence, gen. descant or ‘counterpoint’ accompanying a simple melody (also fig.).
I'm not sure that it specifically means 'written from memory' but I'd say it's pretty close; prick-song specifically means 'from a written source, not from memory' so the initial pricking must've come from memory.
Some more Pepys quotes (from his Diary).
...so I only invited her to come and dine with me on Sunday next, and so away home, and for saving my eyes at my chamber all the evening pricking down some things, and trying some conclusions upon my viall, in order to the inventing a better theory of musique than hath yet been abroad; and I think verily I shall do it.
...he did play beyond anything of that kind that ever I heard in my life; and with great pains he must have obtained it, but with pains that the instrument do not deserve at all; for, at the best, it is mighty barbarous musick. So home and there to my chamber, to prick out my song, “It is Decreed,” intending to have it ready to give Mr. Harris on Thursday, when we meet, for him to sing, believing that he will do it more right than a woman that sings better, unless it were Knepp, which I cannot have opportunity to teach it to.
...and then to Hide Parke, where many coaches, but the dust so great, that it was troublesome, and so by night home, where to my chamber and finished my pricking out of my song for Mr. Harris (“It is decreed”), and ...