Is this really so hard to implement? I had a somewhat similar situation in the Interactive Diagram for Tamerlane II, where 'King Succession' (swapping King with a Prince as a move) is only allowed when you are checkmated. (I suppose the 'only-move exception' here in practice also only occurs when checkmated, as it is hard to imagine you could be stalemated when you still have a Golem.) So I could handle it by some extra code in the section that handles mates once it is detected that you have no legal moves. This code is almost never executed, so there is no slowdown in normal play. It then testst whether succession is enabled, and if so, tries all possible swaps with pieces of the designated successor type.

In an analogous approach you could, in a position without legal moves, generate the illegal Golem captures, search those, and return the score of the best. Since you had already generated them as pseudo-legal moves, and then rejected them in the special legality testing code, you could have the latter code stash the moves somewhere, and let the mate-handling code retrieve them, so you don't really have to generate anything.

Code-wise it is much simpler to just redo the entire node under conditions where the GxG captures would be considered legal. All other moves would be immediate hash hits anyway. Like

Search(stm, alpha, beta, depth, legalGxG)
{
  nrOfMoves = GenLegalMoves(stm, legalGxG); // the second parameter would suppress the special legality test on GxG
  if(nrOfMoves == 0) { // we are mated
    if(!legalGxG) { // well, maybe not really, as we might have rejected a GxG
      return Search(stm, alpha, beta, depth, TRUE); // redo allowing all GxG
    }
    return (incheck? -INF + ply : 0); // stalemate correction
  }
  for(ALL_MOVES) { // normal search
    ...
    score = -Search(!stm, -beta, -alpha, depth-1, FALSE);
    ...
  }
}

You could try to be smart and let the re-search depend on whether there actually were any rejected GxG moves, but since you will virtually never be checkmated, efficiency here is not really relevant. It hurts more that you have to pass an extra parameter to Search. But you could use the e.p. square for that (assuming you pass that, to indicate where e.p. capture is possible). By setting it to the location of the captured G after GxG you could indicate this G is now fair game even when protected.