I wrote some stuff about this in 2021, and tried Live again just now to see if anything changed in behaviour.
{1} If you drag a single chord in section C from Scale to Live, then it shows in the track clip as ‘Scaler-Chords.mid’. This has nothing to do with the file name. You can test this by dragging the mid to the desk top, renaming it it and then dragging it into Live.
{2} If you switch midi markers on in ‘Settings’, imported midi is then titled ‘Scaler-Chords-With-Markers.mid’ ; again, the clip file name has no effect on this, so it appears to be ‘hard programmed’.
{3} With markers off, no chord data is shown in the midi track meta data… With markers on, the meta record has either the chord name or the chords in the sequence.
So a section C thus

has the following meta data
So Scaler does write the name of the chords into a midi file when markers are on.
{4} If you re-export a file which originally had midi markers in (as above), they are not there when exported from Live., So as they were there before import, and gone after import, it is Live that has ignored or deleted them.
{5} So for a controlled test, I cleared state and just put a Cmaj triad into section C
Here’s the hex

We can see that has gone into FF 03
FF 03 11 53 63 61 6C 65 72 2D 43 68 6F 72 64 73 2E 6D 69 64 9E 00 80 30 6E 00 34 6E 00 37 6E 00 FF
Now switch markers on
and we have the Scaler ‘markers’ text into FF03
FF 03 1E 53 63 61 6C 65 72 2D 43 68 6F 72 64 73 2D 57 69 74 68 2D 4D 61 72 6B 65 72 73 2E 6D 69 64 9E 00 80 30 6E 00 34 6E 00 37 6E 00 FF
But we also see before that the ‘Cmaj’. It appears to have gone into FF 06, which the MIDI spec defines as ‘Marker’.
FF 06 05 43 20 6D 61 6A 00 FF
So in the case of Live (I have no idea about Bitwig, but @Bernd or @jamieh might chime in) it seem reasonable to conclude that
{A} Scaler puts fixed text into FF03 with or without the ‘with markets’ annotation
{B} Live uses FF03 to name the imported track clip (not the file name).
{C} Scaler appears to put chord data (either as a single chord or as a sequence) into FF06
{D} Live ignores FF06 on either import or export.
There is one other thing from your examination which you might be able to throw some light on. The MIDI spec (96.1, as amended) defines metadata in FF03 as being for Format 0 (as these clips are) as the “Sequence / Track Name” (as you stated)… There is no mention of it holding chord naming data per se, or is the correct interpretation of “Sequence” here the chord sequence ? This sort of seems at variance with the definition of 'Sequence Number" given against FF00.
There shouldn’t be any problem with it being in there, but will a DAW read that ? It would only be meaningful in track name terms if you exported the chord one at a time from Scaler, and that would cause a problem in a DAW like Live.
For DAWs that do show chord names with Scaler output, they must be picking it up from FF06.