I think the crucial part of information that is missing from your description is that a player who captures the King wins the game. Or, consequently, that one must always move the King out of any opponent check (with the additional rule that it should also not be moved into 'friendly check').
I don't think the rule that "pieces move according to FIDE rules" covers this, as it is unclear to who the King belongs, so that the FIDE rule that you cannot expose your own King to check becomes meaningless. As a reader I would only interpret this rule as applying to the pseudo-legal moves of the pieces.
If my interpretation is correct, then an algorithm for playing this variant would be to remove the King from the board when generating the moves for the player currently on move, and mark all board squares to which moves go. (In this variant these are all pseudo-legal moves of pieces beyond the borderline to empty squares, as without Pawns there are no divergent pieces, and capture of non-royals is forbidden anyway.) If the square on which the King should be got marked, the game terminates as a win. Otherwise we can put a King of the color of the side to move there, (which then would not block any of the generated moves), and generate all its moves that do not go to a marked square. That excludes King moves into friendly check. Whether any of the remaining moves runs into enemy check will be discovered by simply trying out the move (which then result in an immediate loss through the mechanism described above if it would expose the King), as the search algorithm of the AI would normally do.
So it seems it only requires keeping track of the King location (which is different from the normal tracking because it should ignore color, but simpler because there can be only a single piece of the type of interest), and a reasonably straightforward modification of the move generator. This is not hopeless.
I think the crucial part of information that is missing from your description is that a player who captures the King wins the game. Or, consequently, that one must always move the King out of any opponent check (with the additional rule that it should also not be moved into 'friendly check').
I don't think the rule that "pieces move according to FIDE rules" covers this, as it is unclear to who the King belongs, so that the FIDE rule that you cannot expose your own King to check becomes meaningless. As a reader I would only interpret this rule as applying to the pseudo-legal moves of the pieces.
If my interpretation is correct, then an algorithm for playing this variant would be to remove the King from the board when generating the moves for the player currently on move, and mark all board squares to which moves go. (In this variant these are all pseudo-legal moves of pieces beyond the borderline to empty squares, as without Pawns there are no divergent pieces, and capture of non-royals is forbidden anyway.) If the square on which the King should be got marked, the game terminates as a win. Otherwise we can put a King of the color of the side to move there, (which then would not block any of the generated moves), and generate all its moves that do not go to a marked square. That excludes King moves into friendly check. Whether any of the remaining moves runs into enemy check will be discovered by simply trying out the move (which then result in an immediate loss through the mechanism described above if it would expose the King), as the search algorithm of the AI would normally do.
So it seems it only requires keeping track of the King location (which is different from the normal tracking because it should ignore color, but simpler because there can be only a single piece of the type of interest), and a reasonably straightforward modification of the move generator. This is not hopeless.