Just in case there is interest - and because the original question actually comes from a quest to get at music from other angles - here is how "Fuer Elise in F#m" played out for me, with my particular background. Because I tried it that day.
So I'm 95% relative pitch, though some latent pp seemed to develop accidentally via an exercise I was given for something else. Moreover, the only way I had for relating to music, for decades, was via the kind of Solfege I was given in a primary grade one year. That is: major scale with Do as Tonic; minor scale with La as Tonic - modal like the medieaval monks sort of. I used to write music down in my own short hand "m r d" = Mi Re Do. I've had to develop the other side of it, where A is a specific pitch and place on the piano (hard to explain).
Fuer Elise also happens to be a piece I learned early through the hand-me-down book I was given as a child. When my parents moved house and left the piano behind, they gave me a guitar, and I brought the entire piece over to guitar, using the way I hear music to do it. For decades, Fuer Elise has been "Mi Ri Mi Ri Mi Ti Re Do La" (with the minor Tonic being La). This sort of made me hear the relative major at the same time, and it's a messed up system. (V7 is heard as starting on Mi with an altered note.)
So the first thing that happened to me was that darned Mi. In Kodaly Solfege that would have been So. Well, knowing how I relate to music, I decided to focus on the relative major = A major. In either case, the first note in F#m is C#. (which is also Mi in A major).
I then had to build a "keyboard map" of A major or F#m - which black and white keys. Then you're just jumping up and down the degrees of that scale - how I had always perceived music. The first chord is Im or i, the 2nd chord is V7 (but in my messed up Solfege that would have been vi and III (i.e. raised 3rd degree). I've done some work with chords by now: realizing that the 2nd chord is C#7, I switched to the chord playing skills I have gained.
That gave me the first couple of measures of Fuer Elise, and it didn't take long. bit it did take longer.
What I was drawing on was also very close to Puck's system. If music is diatonic, and you can perceive it along degrees of the major and minor scales (which is what C major does for many people by association, aka the "white key scale"), this helps you transpose into all other keys. I had that part.
But I have a weak "keyboard map" - i.e. that muscle memory of A major, F# minor, Db major etc. Anyone wanting to use Puck's system for transposing should have their grasp of major and minor to be solid, including and especially in relationship to the black and white keys of the piano.
It is also good to have a grasp of chords. When you have that Im V7 Im going on, It's easier and faster if you already have chords in your hands, mind's eye, or whatever. There are skills that get used; notation systems by themselves work hand in hand with that set of skills. Which is a conclusion we both came to regarding Puck's demo: that his other skills are part of that demo.