I know that I’m late but all players in the lobby will need the mod menu (common_zombie_patch.ff) file for the menu to work. Same with any other black ops zombies mod.
(This is assuming you are on steam not console.)
Apologies, but I play on PC so I have no experience with console modding. I assume you will need to go into a solo game first then start up your private match, but don’t quote me since I am just guessing based on what you need to do with other menus.