Glass just isn’t going to do clear sharp thin lines like wood. You’ll need to adjust that artwork to fatten the thin parts, and lower your expectations.
The good news is since this project looks like it’s a flat piece, you can use LazerMask and recreate very fine details with an abrasive etch process.
The bad news is, it’s expensive. You’d need an air compressor (>$75) and abrasive blaster tool ($75). And LazerMask itself is expensive. They’ll send you free samples but a roll wide enough for that piece of glass $200-250. So… Figure out some other glass projects to make use of it all. 