the reason you don't find 'real grease' to fit those little guns, is they're made specifically for the white lithium grease, which is too light for chassis lube, better for moving parts in machinery, like conveyors, or electric motors [lithium grease is really oil thickened with lithium and other things] - get yerself a real grease gun and some proper chassis grease, which can also be bearing grease, and comes in a variety of colors and ratings
Bill