OK, so I've been trying to plan what the endgame looks like now, taking into account all the glitches/strategies we've found while the TAS has been under construction. Assuming we're allowing even very low-probability events, and I haven't made a mistake (or if I have, that it's fixable), we can do it in 2003.
Here's the new plan.
Two actions of that plan seem very hard to pull off. Possibly the easier (!) is the long sequence of monster actions at the Vibrating Square level revisit, that involves two 200 teleport fails on the same monster action, and also manipulates a spontaneous polyself immediately afterwards on the following turn boundary. The reason I think this one might be easier is that I have a plan that looks like it might work: fill the level with monsters, traps, and in particular quantum mechanics, and just let mechanic after mechanic teleport us in the hope that at least one of them lands us on the right square (then keep teleporting round until the spontaneous polyself lines up).
The harder one is crossing Air in one monster action and then landing in the perfect place on Fire. A hypercharged conflicted engulfer will move around at random, repeatedly engulfing and unengulfing us. If its random walk takes us to the portal (already unlikely), then we get dropped straight onto Fire, and have to manipulate the portal to be 4 squares North of us (this is closer than the 5 squares West that was previously the best known portal placement). Pulling off something this unlikely probably involves a computer search for an appropriate seed in advance, then planning the rest of the run around it so that the seed gets hit at the right moment.
Another problem with the 2003 turn plan is that for every 4 squares of the random walk, we need to spend 1 turn boundary earlier in the game. This might mean up to 1000 or so turns, each of whose boundaries is just spent hypercharging a monster, which is going to look repetitive (and also kind-of flickery because hypercharging involves changing level) no matter what we do, so I'm not sure it would be that entertaining.
I also have a 2004-turn plan that avoids these problems (using a different and equally impressive strategy for Air, that's slightly slower, and using a manual polyself rather than a turn boundary polyself on D:1). At this point I'm not really sure which to aim for. It might depend on how entertaining we can make hypercharging.
EDIT: The 2004 turn plan is now online
here. dwangoAC and I both have some reservations about the 2003 turn plan, both due to grindiness, and because it seems to be beyond what's reasonably possible to manipulate. So we may have to settle for this one instead.