While there certainly are incompetent interviewers, I think ones that extreme are rare.
See also: overused brainteasers. I remember being asked the 5 litre / 3 litre / get 4 litres problem for my Microsoft intern interview, and honestly told them that I already knew the answer, so I wouldn't be "solving" it, but here it is anyway...
Once I notice that it's not really going well, I try to politely as possible suggest that maybe we adjourn, so as to not waste each others' time. That hasn't really ever backfired, and actually turned one rejection into an offer, but I find it's polite to respect everyones' time.
Engineering is the use of knowledge/science to solve problems, right? So why is asking someone to use knowledge to solve a problem not applicable to the hiring process?
To be clear, this isn't the same question as "why are manhole covers round", or "how would you move a mountain".
If these problems were so reliant on individual problem solving ability, there wouldn't be dozens of books with reams of answers to these questions on the market. These questions test memorization and nothing else.
Correct me if I'm wrong, but a binary search tree is a very simple data structure. It has rules, and just by it's definition, and the definition of "inverted", you should be able to come up with an iterative solution, even if it's just walking the tree and re-creating it.
It may not be the best, but it's a solution, it's a starting point.
But if you're interviewing for a job solving problems, shouldn't you be able to solve problems? Even if they aren't the best solution? Genuinely asking here, as this approach seems to have a 85-90% success rate for me.