You've got a nice list here. While it is not exhaustive, it does seem to cover a lot of ground, and it even puts my suggestion there in the middle. [Nice name for the idea - mind if I use it in my write-up? I freely confess to being naming-deficient.] The problem, of course, becomes just how effective each method is. And, since we're discussing a game, I'd like to add one more constraint, playability, as a separate consideration to look at along with the mathematics of the situation.
The first thing I'd note is that there are mates in 2D on 8x8 that apparently can't be done on a 16x16, for example, although I'd have to ask a couple experts on that sort of thing to give specific examples. They do involve variant chesspieces, but, by definition, 4D pieces are variant, and it's probable that K+Q vs K mates which work on a 4x4x4x4 will not work so easily on a 10x10x10x10, say, which gives more sidestepping room. The 4x4x4x4 is the smallest 4D board which allows more or less standard chess pieces, like the knight, to function as intended. It is exceedingly cramped for a chess board, and prevents running, so facilitating checkmate by the 4D king and queen combination.
However, that leads to the opposite problem: on such a small board, how do you prevent checkmate? A 4D queen is pretty tough; stick her on one of the 16 central locations on a 4x4x4x4, and how many other locations on the board does she not reach? So where do you set her up in the beginning? Someplace where the enemy king is not in check, and the enemy queen is not en prise?
This takes us to the entire initial setup and questions of piece density. What sort of setup is required for fully 4D pieces? Here you should, I think, consider the role of pawns in standard chess. The pawns both protect and block the power pieces, and the 2 sets of them form a pretty stable wall around which the longrange pieces work. What do fully 4D pieces require in the way of pawn protection/blockade for the game to have even such a thing as an opening phase? And what does this do to the playability?
At this point, I'm not convinced that any of the answers, including my own, is adequate. But if you look at the "gladiator kings" idea, it is actually a refinement of "cripple the king". [As an aside, couple that idea with "bare king", and you might be closer to giving achievable victory conditions.] And under the right conditions, it might work very well indeed. Consider that rule for a 3D game, for example, or on a 4x4x6x6, where it should be easier to get to the other side's "king level, because it has more destination squares. Or add a piece - or set of them - that acts as a non-royal king to hold the other side's king in place. Whether it is also held, or free, is a method of "tuning" the piece to aid game balance. Well, anyway, any general idea is subject to refinement [you can see the gladiator idea as providing transient 'fortress' areas, too, for example] in many different directions, I suppose, and that's what I was getting at in this paragraph in my usual roundabout way. [It would probably work better if I didn't leave out so many of the in-between steps…]
I'm looking forward to the next developments.
