it seems very unlikely to actually stay that restricted

I don't know -- my level of uncertainty about what's going to be "likely" or "unlikely" in the event of a limited nuclear exchange is very high :-/

how much influence do they have?

The question at this point will be not "influence" but "capability" -- will they be able to remove Putin and those personally loyal to him at any cost from power? I think it's "likely" but see the previous paragraph :-)