The answer to your question is dependant on at least two things which you have not properly defined. One is why you wish to fight the war (goals) and the other is what actions you consider to be military actions and therefore acts of war (methods). These two dimensions are important because they define a military power's actions and allow us to hopefully differentiate between acts of war and acts of peaceful competition.
So having put those two caveats out there and assuming that there is no defined line between war and peace for our rival powers, the best way to fight a war between two nuclear armed powers which have the capacity to fully annihilate each other in a thermonuclear exchange is to destroy your rival from within. Economic warfare, imposing costs and stresses on your rival's society, arms and technology races to waste scarce resources and impoverish the rival state and its people, proxy wars to drain a more of a rival's wealth, terrorism to do the same and to spread suspicion and revolt against authoritarianism in the rival's society, espionage and sabotage by deniable means to learn vital military secrets and to impose security costs and to damage key infrastructure (both real and virtual), propaganda to weaken the social fabric and societal cohesion of your rival state, exploiting pre-existing social divisions in a rival's society, etc. These are all on the table. So is using international financial and political systems to weaken a rival, interdicting or inhibiting commerce and trade by political and legal (lawfare) means, using national and international media to discredit and demonise a rival, and many other strategies and tactics.
Unfortunately your rival can be expected to do the same to you, so such a blurring of the lines between peace and war creates a forever-pseudowar which not only destroys your rival's society but also transmogrifies your own into a monstrous military dominated and authoritarian leviathan. War or pseudowar requires that concepts like liberty, freedom, free will, free speech, individualism, privacy, free thought, dissent, protest, justice, truth and reality and democracy itself be suppressed while notions like nationalism, sacrifice, obedience to hierarchy, authoritarianism, duty to the collective, suspicion, hate of the other, secrecy, surveillance, subterfuge, militarism, and ultimately arbitrary and largely unanswerable power of the militarised state be promoted. These may be too high a spectrum of costs to pay for winning a pseudo-war with a rival nuclear power. If so then a better strategy is to not fight such pseudowars and if such forever-pseudowar is thrust upon you then to publicise it to all in the international community so that all states can see what is going on rather than secretly waging back such bilateral pseudowar at the rival on your own.
Cheers.
Evilroddy.
