There is no general solution other than brute force. That's not a terribly difficult extension of the halting problem, it just takes more paperwork to deal with the edge cases, but you'll get to that result. The same basic technique works: Your supposed solution to the problem is itself some finite program, and you can feed it the "I halt only if I don't halt" problem too. The difference is that brute force is a solution, because now instead of an infinite sequence of programs you have a bounded set of programs. So whatever concrete "I halt only if I don't halt" you pass to someone within the specified limits, there is definitely some answer, but your technique won't be able to tell what it is short of just running it.

For the same reason the halting problem doesn't even have a good heuristic, neither does this. Unpredictable chaos is not an exceptional case, it is the exponentially-normal case. You have to go the other way, and construct programs deliberately designed to have the ability to tell if they halt. The term for that if you want to learn more about it is "non-Turing complete programming language", sometimes called a "sub-Turing" programming language: https://increment.com/programming-languages/turing-incomplet...

You can read that as "this is how hard it is to construct code that we can make execution guarantees about". That focuses on code that is deliberately constructed to be finite in scope and may be something that can be strictly bounded in memory use or time or both. You'll note if you spend any time working with them how hard they are to work with. That's a reflection of the limits of generalizing any such proofs of time or space of a given program.

If there is a general algorithm that does what you think, we don't even have a clue what it would look like. And we have a lot of clues there can't be any such thing.