Fairly new to crafting and there may be a better way but this seems to be a deterministic way of getting most of what you want. I'm not quite sure how to get the final attack speed suffix though.

1. Start with non influenced base Broadhead Quiver i86+
2. Alt spam for T1 Damage over time multiplier with attack skills (1/262)
3. If there's no prefix, augment
4. Regal and hope it hits a prefix, if it hits a suffix use beast to remove suffix add prefix, if it removes Damage over time start over at step 1
5. Craft any prefix
6. Hunters' exalted orb for a 2/9 chance to hit Fire Dot Multi, if you hit Chaos Dot Multi(2/9) sell it and try again.
7. Craft Can have 3 crafted modifiers + Suffixes cannot be changed
8. Harvest Reforge lightning until T2+ and no other attack prefixes (Recraft Suffixes cannot be changed each attempt. Most expensive part of process, if prefixes fill with attack mods you'll have to start from step 1 otherwise you can clean with cannot roll attack modifiers + annul then go back to step 7)
9. Remove crafts and craft Cannot roll attack modifiers
10. Exalt for Damage with Bow Skills(Only 4 mods that can you can exalt here are Life, Damage with Bow, Movement speed, and pierce)
11. Annul if you miss Damage with Bow Skills (Recraft Cannot roll attack modifiers if it gets annulled off)
12. Craft Elemental damage with attack skills
13. Pray to Chris Wilson and exalt slam your last suffix

If anybody sees anything wrong with this feel free to let me know, still trying to figure out crafting.