I looked up Touch Copy, and it appears to be for transferring music (and other content) from an iPhone/iPad/iPod to the computer, rather than the other way around. Perhaps it also works from computer to iDevice. The website doesn’t mention it (from what I can see).
This use of this kind of software is usually for salvaging content from a phone and getting it back to a computer where it can be used or transferred to new devices. It exists because iTunes usually did this poorly, if it didn’t just erase the iDevice entirely.
If the software did transfer music from the computer to the iPad, it should be in the Music app, and it should not need to stream. If it only copied music from another device to the computer, then the music is probably in iTunes (or another music app) on the computer, and another step will be required.
Since you seem to be able to stream the music, I’m guessing this is what is happening.
1) You managed to add the music files to iTunes on your computer.
2) You have iCloud Music Library turned on for both the computer (iTunes) and on the iPad.
In that case, all you have to do is find the music in Music on the iPad, tap hold on it, and select Download from the pop-up menu. This is done in the Library tab. You can view the music by any of the options at the top left, though it you already have it in a play list you can conveniently download the entire playlist instead of selecting individual songs/albums/authors.
As an extra step go to Settings > Music and make sure Optimization is turned off. If you don’t, and the storage gets low on the iPad it will delete the least played tracks in an attempt to save space.
Other than noticing that the download option is gone from a track/album/playlist, the only way to be certain a song is available offline is to turn of wi-fi and cellular, then see if you can play the song. Or wander somewhere where you don’t have those services.
If I’ve guessed wrong, please let us know more details on how things are set up, so we can guess again.
Good luck.
