Solution 1:

You can use the command /lp user %player% parent add <group> and LuckPerms will add the group on top of the player's existing groups. For example, if the player has rank 1, when the command is run they will be given rank 2, which means they will have both rank 1 and 2.

I'm not sure what you mean by putting a command in an item - do you mean running a command when a player right clicks the item in their hand? If that is what you wish to do, I suggest using the MyCommand plugin. You can run commands through items, blocks and more.

What plugin are you using to give the player a kit? Most plugins like EssentialsX don't clear the player's inventory, that is unusual behavior.