Why? There are downsides to everything but I’m thinking this is the correct action.

Swap "NRA" for "planned parenthood" and "LA" for any conservative-leaning city.

Public contracts aren't supposed to be political. They're supposed to be based on who will do the best job for the best price.

During Civil Rights, the Klan and other hate groups tried to force public disclosure of NAACP roles and disclosure of who supports them. Courts blocked such disclosure to protect peoples right of association because the right of association is a private right.

If your political leanings were forced to be made public you might think twice before supporting someone. That is a chilling effect on speech and the 1A.

This will force all business that want to work with people in LA to just cut any ties with the NRA if they for some reason had them

That is exactly the problem. Forcing people to alter their political practices in order to get a government contract is unconstitutional. You might as well say "Only democrats need apply"