"If deterrence fails". Meaning in your scenario, NATO believes Russia will nuke European cities.
"Lowest possible damage". Referring to the US and Europe, not Russia.
Solution: Preemptive strike on all Russian nuclear forces. Sink the boomers, take out what's left of the Russian early warning networks, command and control, etc. We know where Russia's nuclear stockpile is stored, and we know where their strategic nuclear forces are based.
Since we already know Russia can't even interdict slow-moving drones around Moscow, this shouldn't be too difficult. An warhead entering the atmosphere at Mach 20 and hitting the ground 8 seconds later isn't something Russian defenses can deal with...
Plus there is no reason not to think their nukes are in any better shape than that rest of their garbage military...
There is no such thing as a "limited nuclear war", and your "reading" of this document is completely irrelevant.
If NATO believed what you seem to think is Russian doctrine, the logical course of action is to just destroy Russia before it happens.