Since the oscillating water level in the tuning tube presents an alternation of increased pressure and decreased pressure, I decided to connect the tube to a small tank (just large enough to contain a volume equivalent to the change in volume of fluid between the extremes of the cycle) fitted with a pair of oppositely acting check valves. During contraction, the first check valve would admit fluid to the tank while the second prevented air from being drawn into the tank; and during expansion the first check valve would prevent draining the tank back to the source while the second allowed delivery of the fluid. In this drawing I've added the pump to the basic engine:

A number of people have asked why I've incorporated an air "buffer" between the stirling engine and the pump. It's there to prevent silt and debris from accumulating in the engine itself. It's possible that the check valves and pump tank may need to be flushed from time to time; but I'd prefer that the engine's performance not be affected by the qualities of the fluid being pumped.

If this isn't a concern for a particular application, then it should be possible to eliminate the air buffer and tank altogether.