An attempt at a "good enough" solution for human two-party negotiations

Sure, but what incentive do they have to input their true BATNA?

Just in case: I assume that by "best price" you mean "highest price" rather than "estimated fair price".

No, I mean the price at which that party is indifferent between making the deal and not making the deal.

If we take a general Bayesian framework, I have a distribution on the buyer's best and fair price. It seems to me that most/all nontrivial distributions will incentivize me to lie.

Yeah, and I'm trying to make that difficult for humans to do.

I guess. But I don't know of any real-world transactions where it's expected that people keep their word on something like this

I think there are two points worth raising here:

1. If someone accepts to precommit to the result of this negotiation and then, when the website outputs a price, refuses to honor it then I probably do not want to trade with them anymore. At the least, I would count it as though they agreed to a price and refused to honor it the next day.
2. You only need to keep a solid precomitment yourself to avoid falling prey to the strategy above.

The obvious issue in my eyes is that few people would agree to use this kind of tool anyway, especially for a nontrivial transaction. But in a society (or simply a subculture) that normalizes them it would probably no longer be true that people don't consider their precommitment to the tool as binding.

No trust in the other party is required to use this tool, that's the whole point.