I think I may have found an improvement near the start of the final part, and possibly a new way to finish in turn 2003 using this.
At the start of turn 2000, polymorph into something that flies. Do the usual stuff with getting accepted into the quest, level-teleporting down, and using monster actions to get the Bell of Opening, and now have another quantum mechanic teleport you onto a level teleport trap (there probably are some difficulties here, but I don't have time to figure it out now), and return to the start level.
Now do a zero-time artiblast depolymorph, and plunge into some water. Crawl out onto a land mine on some ice, which explodes. This breaks a lit potion of oil, causing a fiery explosion. The ice you are on is melted, and you drop into the resulting water and crawl out again -- onto another land mine on ice. This also explodes. It breaks a potion of unholy water, triggering lycanthropy, and then puts you into a pit. The explosion of the potion of oil is not done yet, and now it melts this ice, dropping you into water yet again. This time, you have to ditch items before crawling out, because of the difference in carry capacity between your lycanthropic and normal forms and a fine-tuned inventory weight. You drop a boulder; because you are in a pit, you get squished under it, losing rnd(15) HP, which reverts you to normal form. This calls drown again, in something similar to bug C343-193, with a similar result of crawling out twice in a row. Land on the magic portal and a polymorph trap in either order, and turn into a flying form again and leave the Quest; a monster standing on the portal on the other side causes you to be pushed over some water. (Some luck manipulation can be done in the middle of the sequence by placing extra items to be scattered, and/or adjusting the number of items in inventory -- items directly in inventory affect emergency_disrobe, and most items directly in inventory or in containers cause RNG calls in water_damage, even if not damageable, and it is even possible to add RNG calls in only selective drown calls by putting items in greased sacks that lose their grease, possibly nested, or in cursed oilskin sacks.) (Incidentally, the dotrap call left over from the first land mine (line 1161 of trap.c) will cause the second trap to be triggered a second time.) Do another zero-time artiblast depolymorph, drop in, and crawl out onto a level teleport trap, and go to the Valley of the Dead.
As before, level teleport down to the vibrating square and "oLS-cancel it, and read the Book of the Dead, and use the following monster action to turn into an air elemental and enter Moloch's Sanctum. Put on an amulet of life saving, and because this is now the last action of the turn, the problem that earlier arose here is easily solved -- a hypercharged monster can take several actions, acquiring the Amulet of Yendor and taking it up to the level above.
Now entering turn 2001, polymorph into a ceiling-hider, and quaff a cursed potion of gain level, going in the following monster action to level 1 with the Amulet of Yendor in the same way as before. Enter the Plane of Earth by climbing the stairs or by another potion, then hide and drop into the portal as before. Now do a zero-time artiblast depolymorph and kick a monster to hurtle into the portal to the Plane of Fire.
Coming into turn 2002, use a delayed instadeath to end helplessness, and polymorph into a queen bee. Because you were in natural form, you get only two actions. Put on an amulet of life saving and jump into the portal to the Plane of Water.
At the start of turn 2003, have the bubble move to leave you over water. (I'm not sure about this; you mentioned it as a possibility previously, but it appears to me that bubbles carry the player along even if the player is flying or levitating.) Polymorph on your action into a ceiling-hider using command repeat, and drown and 200 teleport fail onto the magic portal -- this will require some egg-hatching to fill up the level, and probably luck manipulation to get the portal at one of the farthest spots so that it does not get covered up by the hatching monsters. Still with three actions left, cast turn undead to fill the Astral Plane, hide and fall onto the altar, and offer the Amulet of Yendor.
