I don't think he would, for a lot of reasons, including that he's afraid of being killed (witness his 20 foot-long tables to prevent COVID/assassination attempts, using his security detail as pretend factory workers for photo ops, having a tight inner-circle, his KGB background, his use of doubles etc.).
But most of all, he's somewhat of a control freak, and using a nuke kicks off a lot of variables that are out of his control. He counted on the West being passive, and he was quite wrong and it's cost him dearly, to the point where Ukraine is not only still kicking 30 months into the main invasion with plenty of Western weapons, but is now hitting places inside of Moscow proper.
Putin was in the KGB in East Germany when Chernobyl happened, and he knows full well how pissed off the West was over the would-be cover-up. He maybe in a bubble of sorts, but he has seen what sanctions are doing to his economy, he knows Kursk was invaded, knows Moscow was hit (and has publicly admitted to both). He knows China isn't willing to send him serious military hardware because of the sanctions. And China has already warned not to use nukes.
Throwing a nuke into the mix - on a tactical level, if it's on the battlefield, it'll kill just as many Russian soldiers, and probably more given the Ukrainians have access to better equipment (and while he may not give a shit about his soldiers, he doesn't have as many as he needs), and it's liable to cause the Ukrainians to move their troops up close to the Russians, daring him to kill a shitload of his own folks. He obviously can't use it in Kursk. He can't use it in Crimea because it'll kill/poison a fuckload of Russian civilians, and because he claims Crimea = Russia, and nuking Russia is not a good look for him.
A battlefield nuke will most likely cause China and India to completely cut ties, and Iran probably won't want to be tied to that albatross, because even a battlefield nuke means the West (and its allies in Asia) are going to come down even harder on anybody still doing business with Russia, to the point of applying the same sanctions to them as are being applied to Russia. China has already been scaling back its financial/trade transactions over fears the US will sanction Chinese banks. China's economy would implode if they got hit with similar sanctions to Russia, and the Chinese government probably can't survive hundreds of millions of Chinese out of work and in the streets (history says governments don't survive that kind of stuff). Even a battlefield nuke is probably going to send some fallout to places that Putin doesn't control, and it's hard to say what would be worse - fallout heading into Russia (which would flip the Russian population out) or out into the Black Sea or Western Europe.
Hitting a city full of civilians would be even worse. Everybody even doing a few bucks in trade with Russia will be sanctioned and cut off from the rest of the world (complete with blockades/embargos), and shit will probably happen in the UN which Russia won't like - even with its veto power, it can't stop a large group of nations independently working together against it.
The fallout is going to go somewhere. If it drifts over Western Europe, hello Article 5 and at bare minimum NATO taking over the airspace over Ukraine. If that shit drifts over Turkey, you can bet that they'll consider it an attack (as will NATO) and will open up the Straits to any and all Western ships and subs.
Most of all, and the TLDR: If he drops a nuke, he becomes a dead man walking, because it would do the opposite of what he wants - rather than making everybody back down, everybody will assume he will do it again, and operation KILL VLADIMIR PUTIN will be in full effect.
TLDR #2: If he was going to do it, he would have done it long before things got to this point. If he thought he could get away with it, he would have already done it - maybe Odessa, or maybe around Kharkiv.