I won't claim to be able to read Putin's mind, but:

Considering that the Russian government continues to sell this invasion to its people as a "special military operation" and bans calling it a "war", what impact would using nuclear weapons have on the internal politics of Russia?

I also invite you to consider the converse: Ukraine is already asking if it made a mistake by giving up its nuclear weapons, and I have heard plausible claims they could construct a nuclear device within a few months… a few months ago.

The US withdrawing all support, may well result in Ukraine making their own nuclear weapons — and attacking Russia with them to make the point, given that Putin seems to have great difficulty understanding that the Ukrainian people don't want him.