I just keep any old brand lithium moly in pretty much everything. That molybdenum disulfide is some slick stuff; works nice.

As for getting grease out of the whole circumference, I wouldn't worry about that too much as long as it pops out a fair bit. It will work its way around as long as its coming through all four arms of each joint.

 

I use a molybdenum disulfide high melting point, waterproof based grease which works fine. It comes in my case in a 25kgs commercial/industrial tin and designed originally for tractors and the like. In all honesty, grease is grease and provided it meets the spec it's good to go.

 

Unicorn tears is still the best but stupid hard to find