You need another 6 tigers, minimum. With just 6 they will include the rainbows as part of the shoal and nip them just like they do to each other. 12 to 15 is the minimum shoal size for tiger barbs.
Tigers are hierarchical - the nipping is part of the way they sort out and maintain their hierarchy. When tiger barbs are first put in the tank they have to sort out the pecking order, which is why they are particularly nippy for the first few days. If you do get more, they will have to go through the process all over again, so expect more squabbling and nipping.
In theory, all the colour morphs of tiger barb have the same temperament, and they just see other tiger barbs, no colour difference.