Hey,
You already tested this with Catalina? I tried to replace all iTunes coming up with "Music", but it still fails unfortunately 
Don't know if apple may have changed something in their links you get the covers from?
Greetings from Zorilor 
Here is my non working script by the way 
if application "Music" is running then
tell application "BetterTouchTool" to set priorTrackId to get_number_variable "trackId"
try
tell application "Music"
try
set crtTrckId to id of current track
set hasCurrentTrack to true
on error
set hasCurrentTrack to false
end try
--this will show the current track information even if the song is paused
--can be changed to display only if playing by removing this: "or hasCurrentTrack is true"
if player state is playing or hasCurrentTrack is true then
if id of current track is not priorTrackId then
tell current track
--track from Library, has local artwork
if exists (every artwork) then
tell artwork 1
set srcBytes to raw data
-- figure out the proper file extension
if format is «class PNG » then
set ext to ".png"
else
set ext to ".jpg"
end if
end tell
set fileName to ((((path to application support folder from user domain) as text) & "BetterTouchTool:" as text) & "music_cover" & ext)
tell application "BetterTouchTool" to set_number_variable "failedToRetrieveArtwork" to 0
-- write to file
set outFile to open for access file fileName with write permission
-- truncate the file
set eof outFile to 0
-- write the image bytes to the file
write srcBytes to outFile
close access outFile
else
--no artwork, will get the info using Music API
set fileName to my getArtworkFromAPI(name, artist, album)
end if
set currentTrackInfo to my updateTrackInfo(id, name, artist, fileName)
end tell
return currentTrackInfo
else
tell application "BetterTouchTool" to set failedToRetrieveArtwork to get_number_variable "failedToRetrieveArtwork"
if (failedToRetrieveArtwork is 0) then
--serve the previously found track info
tell application "BetterTouchTool" to set currentTrackInfo to get_string_variable "currentTrackInfo"
return currentTrackInfo
else
--failed to fetch the artwork, retry
set fileName to my getArtworkFromAPI(name of current track, artist of current track, album of current track)
set currentTrackInfo to my updateTrackInfo(id of current track, name of current track, artist of current track, fileName)
return currentTrackInfo
end if
end if
else
--Music is running but has no current track, hide the widget
return "^Music"
end if
end tell
end try
else
--Music is not running, hide the widget
return "^Music"
end if
on getArtworkFromAPI(name, artist, album)
--sets the artwork to the Music icon as a fallback. Will get overwritten if the real one is found
try
set fileName to (((path to application "Music") as text) & "Contents:Resources:Music.icns")
set searchTerm to name & " " & artist & " " & album
set responseData to do shell script "curl --data-urlencode \"term=" & searchTerm & "\" --data-urlencode \"limit=1\" https://music.apple.com/search"
end try
tell application "JSON Helper"
try
set responseJson to read JSON from responseData
try
set resultCount to resultCount of responseJson
tell application "BetterTouchTool" to set_number_variable "failedToRetrieveArtwork" to 0
on error
tell application "BetterTouchTool" to set_number_variable "failedToRetrieveArtwork" to 1
set resultCount to 0
end try
end try
if (resultCount is not 0) then
set previewUrl to artworkUrl60 of item 1 of results of responseJson
do shell script "curl " & previewUrl & " > $HOME/Library/Application\\ Support/BetterTouchTool/music_cover.jpg"
end if
end tell
if (resultCount is not 0) then
set fileName to ((((path to application support folder from user domain) as text) & "BetterTouchTool:" as text) & "music_cover" & ".jpg")
end if
return fileName
end getArtworkFromAPI
on updateTrackInfo(trackId, artistName, trackName, fileName)
set maxSize to 60
set trackInfo to artistName & " - " & trackName
set currentTrackInfo to "{\"text\":\"" & trackInfo & "\", \"icon_path\":\"" & (POSIX path of fileName as text) & "\"}"
tell application "BetterTouchTool" to set_string_variable "currentTrackInfo" to currentTrackInfo
tell application "BetterTouchTool" to set_number_variable "trackId" to trackId
return currentTrackInfo
end updateTrackInfo
--Thanks to @buseco for the base of the code!