It seems awful transparent, when they do this “false flag” stuff, but I think they are playing to a domestic audience.
I think the US managed to get and share enough intelligence to forestall most effect of the false flag stuff they wanted to use for international justification, leaving them with basically “But NATO...” on that front.
At the same time, the solid international opposition and the sanctions impact make the domestic propaganda front more critical, and there's a lot more that they can do on that front given the degree of control they have over domestic information channels.
The real-time visibility of the domestic propaganda probably further aggravates the international situation, but I think they probably have written that off except direct intervention, which they think, probably correctly, is deterred by their nuclear arsenal.
If they are going to “find” evidence to support Putin’s claim of an offensive Ukrainian nuclear weapons program supported by the US, presumably it will be at one of Ukraine’s actual nuclear reactors.
While it's not a hard limit on propaganda, actually having physical control makes a story about such a find more plausible.