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Actually of course White always should succeed, even against Maharaja (Knight + Queen) piece-type too, several times justified in Chess Variant Page, recently at
 http://www.chessvariants.org/index/displaycomment.php?commentid=24978. Variantists I was designing with once settled on that larger board is enough for the Maharaja. That may not be correct, and offhand now it is unclear which square size reverses the spoiler. 12, 11, or just 10? The cook can mean either is sure to win or actually there is unavoidable draw by good-sense play, according to board size. '8x8' is thus cooked whether Queen or Amazon by win of f.i.d.e force. That one single size of many possibilities has about the only decided outcome to date. On theoretical large enough, Maharaja-Amazon at least Draws, even against nine Queens from 8 promotees. Even though they have to move and keep moving, cannot RNBQKBNR always stay connected no matter the size? In time, that is, since neither Rook initially is ''connected'' in full sense of ''protected.'' And then Maharaja cannot pick off so much as a single Pawn. However, not so. That connected protection does not necessarily close every line of attack on King when the board outsizes the array, and not every square is reached by the 16 pieces/pawns. A piece interposing a check for example may unguard so much as a Pawn.  So instead by 12x12 does Maharaja take Pawn, or Rook, if just by starting checks on King from behind the lines, that have to be advancing if any Pawns move at all, giving freer rein? Then imagine Maharaja, who is hard royalty to trap, can capture up to the whole force before checkmating. Which is it for which sizes? Is there intermediate size where Maharaja wins, or else there is real competitive Chess with no apparent advantage? The latter is supposed to be realm of playable CVs. And on board far enlarged, does Maharajah not just Draw but Win for sure? There are experiments too with other Armies against Maharaja, one example 10-wide with Carrera pieces RN and BN. Surprisingly on 12x12, 14x14 etc. it takes study to settle what one solitary RBN-Maharaja does against Queen, Centaur, Champion and the entire f.i.d.e.-counterpart other 17 pieces and pawns. [All above is assuming the classic Maharaja and Sepoys version of the objectives.]

The key difference between this variant and Maharaja and the Sepoys is that
the Maharaja need only check the King -- not checkmate it -- to win the
game, and it doesn't matter if it puts itself into danger doing so.
Because of this, the Kamikaze Queen is much more powerful than the Q+K
compound.

Now, I'm not anywhere near a master Chess player, but I played it against
myself a few times. Most of the White openings I've seen are Fool's Mates
that open the King up to be checked in one or two turns, thus losing the
game.

I suspect that Red is intended to be played by a very weak player; with
strong players I expect the kamikaze should have a major advantage, though
with some deep level of perfect play, I suppose it might be possible for
White to win, but I haven't seen anything like that yet myself. If anyone
can prove that Red doesn't have a forced win, that'd be much
appreciated.

I think it's really more of a puzzle than a game, in any case.

Proving that red does not have a forced win would be equivalent to proving that white has either a forced win or a forced draw.  Even if that's true, it would be quite a lot of work to prove.  However, even if red has a forced win, the game might still be nontrivial if it is sufficiently long and complicated.  (Remember we still don't know for sure whether either player has a forced win in orthodox Chess.)

I'm pretty sure red can win within 3 moves if white opens with anything other than e3.  Most openings allow 1...Qe8, 2...Qxe+.  If 1.e4 then Qd4 threatens Qxd2+ or Qxf2+ and white can't block both.  If 1.d3 or 1.d4 (so the Bishop can jump in front of e pawn), then 1...Qh5 2.g4 Qxg4 and 3...Qxe2+ can only be stopped by 3.d3 Qh4+.

I'm not seeing any short forced win if white opens with e3, though there are several sequences red can try where white has to make exactly the right counter several moves in a row.

A computer could probably tell us by exhaustive computation whether red has a SHORT forced win, at least (say, within 10-20 moves).  Of course, that might be construed as 'ruining' the game.

I played some games in the e3 line, and it does look like a solid defense
-- I can't be sure, but it looks like an eventual forced win for White to
me.

It's definitely an unforgiving game, though, because if you make one
mistake as White, you're finished.

Building an unbreachable defense is probably a part of any winning strategy for white, but surviving long enough to build it looks like the hard part.  After what opening is it possibly safe to castle?