Not sure what fuse they provide with the harness. What color is it? It should also say on top of the fuse if you can read it, but each fuse is coded, the color corresponds to the amount of amps it'll handle before blowing.
From what I've been hearing slim ballasts tend to run hotter and therefore pull more amperage, but don't just replace the fuse with a higher amp fuse because you could cause more damage.
This can help figure out which fuse it is:
• 5 amp - orange color
• 10 amp - red color
• 15 amp - blue color
• 20 amp - yellow color
• 25 amp - clear color
• 30 amp - green color
I've dealt with many melted fuses, You may want to consider cutting the holder out of the harness and putting in a new fuse holder with fuse.