Are there little seat belts or other restraints for the organ-containing coolers kept in cockpits?
"between the pilots" depends on the aircraft. On larger commercial aircraft and even some fairly small planes the area between the seats is occupied by the FMS and throttle units and there really isn't anywhere to put a cooler as seen here: 



(source)

A lot of aircraft (bigger than the PC-12 mentioned) have jump seats in the cockpit. If unoccupied these seats have a set of restraints that could theoretically hold a cooler depending on size. 

You can find the legality of putting stuff on the floor in the cockpit in answer to this question.

For short distances organs move in helicopters and for longer distances they may move in a private jet presumable in the back with a courier. I would not jump to the conclusion that coolers in the cockpit are organs. Ive seen a few organs loaded into lear 35's at my local airfield.



For the record, if you ever see a cooler in my cockpit its more than likely a ham and cheese sandwich.