Note, this conclusion in contingent on my conclusion that middle C is always ~262Hz.
I wouldn't say that soundfont files are named incorrectly, but i would say that they are using the musical theory standard for labeling the notes, with C4 being middle C. That pitch, middle C, should always be displayed in the first ledger line below the treble staff.
The issue isn't that the label for the octave is wrong, because it can change depending on what standard you are using, but that where it shows up on the staff is incorrect.
Therefore, the range of pitches that this picker provides would need to be changed. However, as you've noted that would impact any saved user tune.
The only way I can imagine fixing it, without changing the octave at which previous users saved their tunes at would be:
Option A: Use the Alto (aka viola or C) cleft instead of treble clef
- Using this cleft, the position of all the notes currently would be correct because middle C is the middle line of the staff
- This is not a common clef, only used for certain instruments like viola and alto trombone
- note names would remain unchanged
Option B: Support an additional octave
- keep the note names the same, but add support for an additional octave higher
- would need to add mp3 files for the additional octave
- the display for the note picker and the game on the left would need to support displaying an octave below middle C, in order to display C3 correctly relative to the staff.
If we weren't concerned with loading saved user tunes, then I would say that the easiest solution would be to just play the notes an octave higher. We would need to rename the files downloaded from soundfonts and would also mean we were would not be using the musical theory standard of note naming (which doesn't mean its not correct, but means we would be following the same pattern as some other music apps).
We could opt to not fix it, but I'd consider it's behaviour to be incorrect.