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There is no 'paradox' here; just an ambiguity of the original rules, which can only be solved by more detailed specification on what constitutes a check. The same issue already occurs in orthodox Chess. In particular the question whether pieces that are pinned on their own King have there checking power subverted, or whether a King might step next to the opponent's King when he is protected. This is then solved through introduction of the concept 'pseudo-legal move', and stipulating that it is not allowed to expose your King to pseudo-legal capture. When one would have specified instead that a move is illegal when it exposes the player's own King to legal capture, this would be a circular definition. But that still doesn't make it meaningless; it just means it must be applied recursively. And fortunately that recursion always terminates, as there are only two Kings to capture. So the question really boils down to: in a contiguous sequence of questionable moves, will the first violation or the last violation be decisive.

Note that in Tai Shogi the opposit holds for the Emperor as in orthodox Chess for the King: Emperor's are allowed to move into each other's range, provided they are protected. (It must be, as the Emperor's range is the entire board!)

As to 'Yavalath Chess': note that the distinction between 'cannot' and 'loses if' is only relevant for determining whether stalemate is a draw or a loss, and has no bearing on the issue of what 'check' means.