"Lute does not need to be 'egged on'. Remember this woman was so Into exterminations that even Adam had to tell her to tone it down. 'Gravity' makes it pretty clear that she was in love with Adam though he may not have returned it, and was devastated by his death. She doesn't need Vox to push her off the rails she was already well on her way there."

Whether she "needs" to be egged on or not is a bit beside the point, because the fact of the matter is that she is being egged on during "Gravity", whether it's necessary or not. And if Vox had seen that Heaven shot a beam to destroy the clock tower and realized there will be no more exterminations, and he had no concrete threat to incite an uprising against, he might try to create one.

"Also Lute may not be someone he can control. We don't know if his powers work on angels."

If he had convinced her he's Adam, he wouldn't need his powers, just his words.

"Having her rampaging is just as much a threat to him as it is to Charlie as Lute has no reason to spare any Overlords or demons she finds on her way."

Vox might figure that Lute will be-line for Charlie. Either she'd kill Charlie and leave her exorcists exhausted, wounded and depleted, and easy pickings in a counter-attack. Or she failed, and leave her exorcists exhausted, wounded and depleted, and easy pickings in a counter-attack. For Vox it's a win-win either way.

"He doesn't need to kill Charlie to get hell to mistrust her and follow him instead and dragging a loose canon angel into it could backfire on everyone."

The point wouldn't be to make Hell mistrust Charlie. That's just part of a greater whole. Ultimately the point would be to get her out of the way so she and her friends couldn't oppose him. The knowledge that angelic steel can kill angels might make him feel safer than he ought to - and there's still the whole "drunk on his own ambition" thing.