The method by which the Crown assents to petitions is via signing bills which the elected government of the day and the House of Lords have voted to bring into law.

In theory, the Crown could refuse to do so, or attempt to bypass Parliament by issuing a royal decree. In practice they'd find such efforts ignored at best, more likely their constitutional role far further curtailed (thence leading to a rapid diminution of their ceremonial duties and privileges, if not outright abolition).

The situations under which the Crown could flex their power and have a reasonable chance of survival are extreme. Even here, they'd de facto be acting in tandem with the populace to defend against a coup and preserve (or restore) Parliament as the source of law within a representative democracy. Whether or not they'd do so in a very clear cut scenario is moot; in the dense fug of populism as a cover to usher in authoritarianism, absolutely not, let alone the drear realities of a clumsily formed electoral system chafing and fraying in a complex world. Against that, neglecting to intervene in defence of the realm from clear attack could also prove fatal to the Crown, albeit far less hazardous to the monarch themselves & their family, having greater opportunity to go into exile beforehand.

Either way, they'd have to be confident that a large majority of the Armed Services, Police and other key institutions, all members of which swear an oath of loyalty to the Crown, were up for obeying orders issued by the Crown against a hostile takeover of democratic institutions.

Whether as a temporary measure to defend the nation in an emergency, or any other cause including those which are malign, a monarch acting as a supreme leader would likely have to use a good deal of their personal wealth to fund their activities. In this they can quite easily outspend many actors.

Broadly, it may be more effective for the Crown or the monarch in their own right to discreetly support an array of resistance groups, than wield regnal power with overt grandeur in the face of grubby onslaught. Meantime... we all best be glad that is vanishingly unlikely that the current monarch or his heir would decide to avail of a severe crisis as an opportunity to seize absolute power.