This is a post-apocalyptic setting where the apocalypse had an apocalypse. The King's Element is immensely valuable and possibly more valuable than a political loss, because political power grows out of the barrel of your singularity gun, and building singularity guns is pretty goddamn hard right now. Arachne literally doesn't even have any idea where to start building something similar, and she's a pretty powerful potential force. More importantly, if Hidalgo does have something useful which can result in a reversal (which he implicitly does, because there is little reason for him to lie, given how the King wants him actually destroyed, and his base of political power broken), this is probably the best scenario to deploy it in - the enemy believes they have the upper hand and is more likely to expose themselves.
Also, if we want to talk pragmatism, Hidalgo's implicitly offering to blow some extremely powerful trump card of his. A trump card that he might well be able to use on us later. Getting our frenemy to blow his advantages here and now is going to make him more of a known threat and danger if and when we have to face him again. Meanwhile, not expending the wild card means we still have the wild card, and the Chance Illusion is throwing around some ludicrously powerful bullshit so they're probably a cut above the other sorts of foes we've been facing. There will definitely be opportunities for the wildcard if we don't expend it.