How can I complete 'The Fated Four' as a bow user?

Bows have a lot of nuance to them, and require particular care in order to bring out their highest damage output:

• Always make sure you're in critical distance, where the screen shakes on impact
• Try to always reach a level 3 charge, and follow up with a Power Shot whenever you can
  • Related, Focus is incredibly good as a Bow user, probably the single best skill for it
• Use Power Coatings, and bring combines for more if you run out
• If having trouble managing stamina, bring Dash Juice
• Rapid shots are significantly easier to use and hit weakspots with, they'll usually deal more damage than Spread shots
• Since your most used shot will be the level 3 charge, make sure the Bow you're using has a powerful level 3 charge shot. Some Bows may only have a Rapid 1 shot on their level 3 charge, lowering damage output.
• Learn weakspots and always aim for them

If you're still having trouble, nothing beats good old practice. Hunt the four monsters separately until you know everything there is to know about them, then try The Fated Four again.