They don't need some new bill to cause chaos, particularly with the way the ATF has behaved lately.
All they have to do is make another rule classifying 'most' handguns as SBR's based on the GCA's definition of handgun which defines it, just roughly here, as being designed to be fired with one hand.
It doesn't say 'can' be or 'is' fired from one hand, it says designed to be. And most modern handguns have distinctive design features to support two handed operation. Most marketing materials will show the handgun being operated with two hands. Hell, most people with half a brain, even back in '68, would have said a handgun should be controlled by both hands, when possible, for safer operation.
So... yeah, they don't need a bill. The law actually supports what they want to do should they so choose. Unlike the pistol brace fiasco and the whole frames and receivers thing -- they'd actually have pretty solid ground on that one.
All of it is just forcing the courts to the inevitable path of dismantling the vast majority (basically all) of gun control. Really, we're a lot further down that road than I expected. A whole lot. That's all thank to overzealous leftist states and a complete freak-out by the ATF/Biden-admin. If they'd left well-enough alone, it would have taken us five years to get where we're at, in terms of litigation and the application of Bruen. They keep this shit up and the NFA/GCA will either be gone or affirmed before the next president is elected.