Can I somehow control a PSU-powered fan using a 4-pin fan output on the motherboard?
The motherboard is able to control 4-wire fans using PWM, and 3-wire
  fans using DC voltage level.


I did not see anything about 3-wire fan support on that page, but if user manual says so then you don't need anything else but correct wiring on 3-pin plug. The PWM control pin will not be used, and the taho output from 3-pin fan is compatible with tach input on 4-pin fan connector. Furthermore, 4-pin connector is usually keyed in a way that forces correct connection of 3-pin plug.

Note, that if motherboard does indeed support both types of fans then most likely you have to configure fan type somewhere in BIOS.


  I originally plugged it into a (known working) fan terminal but it of
  course would not turn


There is nothing "of course" about this. The fan should have been working at full speed all the time and motherboard should have been able to read its RPM. That's, of course, if it is configured for 4-pin.

From the link to a fan it seems it comes without a connector. So, check your wiring first.

UPDATE:

If your problem turns out to be insufficient motherboard output wattage, then one DIY solution would be to add OpAmp in voltage follower configuration, supplied directly from PCU and controlled by voltage output from motherboard connector. Depending on motherboard driver's design you might need to add a load resistor on the control line to simulate fan motor.