Do multiple ground bus bars need to be tied on subpanel?
Depends, are we dealing with NEC, or Harper (that's me)'s Law? 

NEC: The screws attaching the ground bar to the panel are fine if they are 32 thread pitch or finer (e.g. 8-32, 10-32, etc.) 

I would also run a ground wire with the thickest wire you have on hand.