I think you are making too much of this. For one, QS is never omniscient. The stand-pat assumption that the best non-capture will leave the evaluation approximately as it is, is just an assumption that is often false. There could be forks, or skewers. There could even be mate in one. Minimax turns out to be very resistant to random perturbation of a reasonably small fraction of the scores in the leaves; you must be very unlucky for such a perturbed score to affect the root score and move.

You actually defined different classes of legality, and why stop at three? Generalizing this you could have an arbitrary number of such classes. I would prefer to call these 'priority classes'. You must play a move from the highest non-empty class.

What to do is just a matter of statistics. How are the moves typically distributed over the classes, and how are the moves that typically change the evaluation distributed. In Suicide captures are class 1, small in number, non-captures class 2, large in number. The situation that there would be a class-1 move we do not want to consider in QS, which would outlaw all moves we do want to consider, because these are all class 2, never can occur. Same with Jumping Chess; the unconsidered moves are by definition class 2. Of course QS is still affected, because if there are moves to be considered, there will be no stand pat.

In Golem Chess typically all moves are class 1, and the occasional move that can be class 2 is even rare amongst captures. The logical thing to do, considering these statistics, is assume there will be class-1 moves amongst the non-captures. Unless you have reason to suspect there might be none. You would only be interested in knowing that if you did not find any class-1 captures, and when a lot of captures turned out to be illegal you could suspect that might be true for non-captures as well. (You know from the rules there cannot be any class-2 non-captures in Golem Chess.) Or you could simply know that you are in check, and that this is the reason for the large fraction of illegal moves. Especially if it is a contact check, for which the only legal non-captures can be King moves. But then you would probably extend even in QS, and the problem disappears. Otherwise you assume there will be legal non-captures, and guess their score as the current evaluation, to stand pat. It is what you always do when not in check, and the fact that there was only a capture that was discarded because it wasn't class 1 doesn't do more damage than that there occasionally is a class-1 non-capture that would score far higher than the current evaluation.

Note that the engine is very well capable of determining the distribution of moves over the various priority classes from nodes in the full-width part of the search. In particular you could make it keep track of how often there are no class-1 non-captures in the case there are no class-1 captures, and use that to decide whether to do a class-1 stand pat in QS. If there are more priority classes you would want to know what typically the highest-class non-capture is, given the highest-class capture that is available. If that is lower or equal to that of the highest-class capture, the capture can be searched, and a stand pat of the assumed highest-class non-capture can be tried only if that belongs in the same class.