NICS is operated by the FBI.
If for whatever reason NICS were to be shut-down (read defunded/no funds), you had better believe that an immediate filing for an Injunction would be filed in D.C. Even without Ezell as a persuasive precedent, it wouldn't take long for a Permanent Injunction to be issued to enjoin enforcement of 18 U.S.C. § 922(t), the Brady NICS check, as long as it remained, now or in the future, incapable of functioning.
The NICS check itself would not be ruled unconstitutional. It is a very narrowly tailored law that is directly related to a compelling government interest. This law can and will withstand the strictest judicial scrutiny. It is only when/if it fails to function, by government action/inaction that it can then be enjoined, as it would then constitute a complete ban on guns. Something that is absolutely precluded by Heller.