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There's not, because it's shitty and nonsensical to compare an entire gender to a dangerous animal in the first place. The bear analogy did not actually work, and no equivalent analogy for the entire female gender will work either.

I feel like you've locked in too much on the gender binary here and are maybe missing the point of the "would you rather" to some extent. It isn't really about the individual answers to the question (though in aggregate they are also telling.)

The premise revealed by the question itself is that the statistical danger of encountering a random bear in the woods is comparable to the danger of encountering a random man alone on the street. The aggregate answer just feeds into this premise: many woman would take their chances with the bear. This is a surprising outcome, and is meant to make ~~people~~ mainly men question why they feel surprised by this result. To self-analyze and maybe improve on that is the goal.

So for the "opposite scenario" you're asking if there's some surprising situation where an "average" man would prefer to be in than encountering a random woman. I have a strong feeling that due to the general balance of power in society, the answers to this question will tend to still be empowering of men and dismissive of women. I think the obvious joke here is something like "who'd you rather find at home after staying out too late drinking: your wife or a feral racoon?"

The sharing emotions one you use is better but still kind of in the same genre: it isn't pointing out some truth about women, but making a different sweeping critique of men. I thought of "which would you rather: therapy or fight club" but that's the same joke, and is still focused on critiquing men.

I'm having real trouble thinking of a broad problem with (USA specifically) women that affects men and is compatible with this joke format.

“would you rather share your emotions with a woman you know or a tree in the woods”, but that doesn’t really feel like it’s an equal question in my head.

Why not? The premise of the bear vs the man is a physical dilemma, which is usually how a male aggressor attacks a female victim. The premise of sharing your emotions with a tree vs a woman is an emotional dilemma, which is usually how a female attacks a male victim.

The tree vs woman was instantly understandable to me, because I've been manipulated and harassed by multiple women in my life, after opening up to them and being vulnerable, mostly with romantic partners. My male friends didn't do this in nearly the same frequency. That said, it's a problem of sampling bias. I've never had a romantic relationship with a man. Maybe they would physically attack me. Maybe they would emotionally attack me.

As inifinitesunrise said, it's kind of nonsensical. I think the root of the question is "would you rather be vulnerable again or not?" but with heteronormativity or sexual dimorphism. Honestly, no. I'll never be that vulnerable again. It's backfired every time in my life. Every. Time. If my experience is universal, which I hope it's not, then it's just that humans are flawed and terrible. (I'm slowly turning into the old man in the woods with only a dog breaking the solitude.)

Would you sign power of attorney over to a hornet?