Judges probably will not like the definition fight, but with such poor ground the negative will have to put some ink on that part of the flow. Run a resolutional analysis declaring arguments irrelevant if they deal with effects of possession, and not with possession itself. Afterward, focus on winning either a deontological or utilitarian critieria. That way, even if you lose the definition fight, you can fall back on winning your framework, and hopefully the debate round. If you wish to run deontology, you should focus on a social contract sort of argument.
Governments have an obligation to provide a nuclear umbrella for the security of their people.
If you wish to run utilitarianism, you will focus more on the deterrent effect of nuclear weapon possession. If they claim nuclear possession allows bullying, turn the tables on them by arguing that bullying is only possible in a world where nuclear weapons are not allowed for everyone. Affirmatives need to make two things very clear in the debate. First, public office is a very special circumstance concerning the issue of privacy. Average citizens are not going to be pilloried if we scrutinize the private life of a candidate for public office.