But at the same time, should Ukraine start lobbing (western supplied) missiles into Russian towns and cities as a tit-for-tat response, they would immediately lose the moral ground they have thus far held in the eyes of the world. I could imagine NATO taking a dim view of this as well. Its one thing for the West to supply weapons to help Ukrainians defend their homeland and drive the invading Russian forces back out from within their borders. But start shelling innocent Russian civilians in their towns - lowering themselves to the level of Putin - would be a dangerous escalation, and would not help them in the propaganda war.
You say Ukrainians shelling Russian towns would make Putin even more unpopular in Russia. I think it could have the reverse effect, and end up giving him a mandate to do something more drastic in response.
As I understand Ukraine are under strict orders not to attack Russia and Belarus territory.