REMEMBER, THESE BUILDS CAME FROM BOREDOM AND LACK OF SLEEP.
THEY ARE NOT RECOMMENDED.
You can try them for fun, and post your results in the comments (I'd prefer if this was done in a bot game, custom game, or a full team, unless you enjoy pissing people off.)
Only the two ad carry builds are really considered viable.

Ad with BoTRK deals 6429 total damage, assuming every shot of the culling hits, and he got off four basic attacks, applying his passive on each, and applying triforce. This also assumes he got a critical strike on 67.8% of his 4 normal basic attacks, and 67.8% of his passive basic attacks.

Ad with bt deals 6714 damage, assuming all 28 shots from the culling hit, and he fires off four basic attacks, applying his passive on each, and applying triforce. This also assumes he got a critical strike on 67.8% of his 4 normal basic attacks, and 67.8% of his passive basic attacks.

Ad with fully stacked bloodthirster deals 7218 damage, assuming all 28 culling shots hit, and he fires off four basic attacks, each applying his passive and triforce. This also assumes he got a critical strike on 67.8% of his 4 normal basic attacks, and 67.8% of his passive basic attacks.

AP deals 9369 total damage, assuming every shot of the culling hit, and he got off four basic attacks, applying his passive on each, and applying lich bane. With runaans, this fires 24 bolts, 12 dealing 25% of his ad, 8 dealing 50% of his ad, and 4 dealing 100%. All apply nashor's tooth normally. Note, this is to three people, single target damage is significantly reduced.

Single target ap deals a total of 6774 total damage, assuming every single shot of the culling hit, and he got off four basic attacks, each applying his passive and lich bane.

Note: These do not include the burn from liandries, or the passive of BoTRK, due to their scaling based on health.

Using only basic attacks, for four basic attacks, each of the builds do the following in damage:
Bork: 1217
Bthirster (no stacks): 1458
Bthirster (full stacks): 1624
AP (multi target): 1852
AP (single target): 724

Using only abilities, each of the builds do the following in damage (as always, assuming every shot from the culling hit):
BoTRK: 3805
Bt with no stacks: 3727
Bt with full stacks: 3991
AP: 4249

Of course, ap is more susceptible to dying, but it does a lot more damage, sustained in teamfights, and burst in a 1v1. If we were to take out the ga from the ad builds and replace it with a last whisper...
The total damage from 4 auto attacks would increase by: 161
And damage from abilities by: 2802 (and 2452 on bloodthirster build)

Note: this is damage dealt before penetration and resistances are put into account.

Damage comparison from full offensive builds:

Basic attacks:
BoTRK: 1378
Bloodthirster (0stacks): 1619
Bloodthirster (full stacks): 1785
AP (multi-target): 1852
AP (single target): 724

Abilities:
BoTRK: 6607
Bloodthirster (0stacks): 6179
Bloodthirster (full stacks): 6443
AP:4249

Abilities and basic attacks (no passive or sheen procs):
BoTrk: 7985
Bloodthirster (0stacks): 7798
Bt (full stacks): 8228
Multi target ap: 6101
Single target ap: 4973

(with passive and sheen procs):
BoTRK: 9394
Bt no stacks: 9327
Bt full stacks: 9837
Multi target ap: 9369
Single target ap: 6774

As you can tell from this, AP deals almost as much damage as AD when targeting 3 or more people, but falls off massively in single target situations. If ad gets a guardian angel instead of a last whisper, ap completely destroys ad in the damage area, but lacks the survivability.

Basically, ap isn't actually weak on Lucian, its just not the strongest.
(this math does not include any bonuses from masteries, other than the bonus 5% ap on hit in the ap build.)