I'll start with "I don't know & I've never tried it."
Famously, Dave Stewart & Annie Lennox used to do her vocals on a hand-held SM58 in the control room with the phase of one monitor inverted, theoretically creating a null-space in the middle of the room.
All well & good, but I've actually seen video of her putting vox down & she doesn't stay anywhere near the centre of the room - so I'd call that 'hype'.
On the other hand, I've also recorded vox with a hand-held 58, in the control room without inverting the speaker phase.
It's not something I would have volunteered to do, but I needed the vocal performance on video at the same time, as an actual to-camera performance, so standard techniques were out.
U87 & cans were not on the menu.
Results were remarkably good. The 58's off-axis rejection, combined with the sheer proximity of the vocal compared to the monitors left me with a perfectly usable vocal & barely any spill at all.
The vocalist was free to move as he liked, & although didn't wander so far as Annie Lennox in the video I saw, it probably allows that her method worked for the same reasons.
So, hand-held 58s & you're good to go.
I'm not sure how far you'd get with an omni & everybody gathered round at a yard from the mic. A cardioid with its back to the monitor, maybe.
I think the idea of feeding an inverted signal into this is going to be fraught with difficulty. I don't think the frequencies will be time-aligned well enough to be able to get full cancellation. You are undoubtedly going to be picking up stray refections from the walls etc too.
You might get some joy if you can gradually rotate the phase to find a sweet spot, rather than merely flipping it 180° & I'd certainly say it would be worth a quick test - before the vocalists even get there - just in case it does work... but I wouldn't hold high hopes for it.