Let's say we were unlucky and have a crippled stack with only 4BB.

In a big blind ante tourney, wouldn't the math suggest going allin UTG with any two cards?
Because it's the last chance to have somebody else pay the big blind + ante (which is 50% of our stack)

Kinda. I still think I would fold bottom of range 23o, 48o, 59o etc. might stick it in with 56o or 75o though. Idk I would fold pure bottom of range that has bad equity against almost all hands.

You have 2 bb next hand and then 1.5bb and can wait for your spot. At 4bb, I wouldn’t worry too much about your tournament life though haha. 4bb time is on the the next one time

I agree with above. Grip and rip all but the stone bottom. This isn’t worth much thought in your average mtt spot. On a satellite bubble or FT spot with other extreme short stacks that could certainly warrant some strategy.

Without a BB ante, the strategy was to gii on my ou bb rather than earlier. With a BB ante, you are better off gii before the bb because you can't win multiples of the ante money.

I wouldn't put it in with anything since you're so likely to get called. I'd want at least a reasonable chance to win. A high card, or something suited or at least medium-connected.

4BB is enough that even if you have to fold this hand and get it in on your BB, you can still win the hand and have a shove stack with fold equity. (Plus, you'll get to see the action before you have to decide, in case it gets particularly hairy and you don't have a hand.) At that point you're getting 2.75:1 to get in your last 2BB after posting; you only need 26.67% equity to break even, so you should basically be getting it in with any two cards as long as it's heads-up. 7.5BB (or 7 if you fold your BB) is enough to get folds when you open shove.

With 3BB I'm more inclined to shove because even if I win out of my BB I'm still only at 5.5BB. But I'll still fold the absolute worst hands and just take my chances the next hand.

I feel like surely someone has come up with +cEV chip ranges that take the BB ante into account by now.

Certainly we're trying to find something to get allin before the BB arrives.
But let's just say we don't and are evaluating Allin blind UTG vs from the BB.

Here is the math:
UTG we'd be risking 4.0 to (most likey) win 9.5. EQuity required = 42%
From the BB we'd be risking 2.0 to win 7.5. EQuity required = 27%.
So at first glance it would seem that waiting for BB is better.

But, let's assume we have crap equity either way (say 30%)
EV shove UTG = 9.5 x 30% = 2.85BB
EV call BB = 7.5 x 30% = 2.25BB

Our bust rate would be the same at 70%.
But going in blind UTG is mathematically superior to waiting one hand for the BB right??

If the equities are going to be the same either way, then yes, it's more beneficial to shove UTG than get it in from the BB. But the thing is, you don't know that, and that's part of the advantage of waiting.

UTG you're shoving blind into 8 people (or 7, or however many). From the BB you already have a picture of everyone's actions preflop, so you have a better idea of what range you're up against and what your equity will be against that. You can get away from trash if there's a lot of action preflop, whereas if you commit UTG you have no options in that situation. Part of the advantage of waiting is that you have a better idea of what your equity will be when you do get it in.

(And, remember, we're really talking about the most borderline hands here; it's not like we're passing T9s UTG to wait for the BB. It's more like, do we get Q2o or J4o from UTG and hope our high card holds or makes top pair vs. whatever happens behind us, or do we wait to see if we can get it in heads-up from the BB?)

Not Really... Starting from 4BB, after we put in the BB + ante, we'll have 2BB leftover (& I hope we're NEVER folding that)
So either way we'd be getting allin with a random hand vs the best out of 8 other hands.

Both Not Great. But which is the least not great?

These calculations all seem to ignore one fact (unless I missed it).

If you shove from UTG, you still have to play and pay the BB/Ante.

The comparison needs to be

1. Shove UTG, survive, play the next hand (pretty easy to calculate our -ev there, probably something around 1.2BB)
2. Fold UTG, play our BB.

It is only once we have played both of these hands (assuming we survive) that we can compare the two actions, once we reach the SB