I’m afraid you can’t adjust the position of individual lines of lyrics on percussion kits, but you should at least find that they are not colliding with the staff or the notes.
The issue I was having has been resolved. When I used multiple lyric lines for different stickings, they weren’t being vertically distributed and instead were all on one line. They were also too high overlapping the notes.
As @dspreadbury says, you can’t adjust the vertical position of the lines individually. However, you can adjust the spacing between the staff and all lyric lines as a whole in Engraving Options > Lyrics > Vertical Position. I think you may need Pro for this option though.