A shorter prop should give lower climb rate, rather than higher, all other things being equal. The reason for this is that the longer prop will have greater mass flow and so be more efficient. The same is true for multi-blade props because for a given diameter, a three-blade prop will have 50% more mass flow (Doesn't this assume equal blades area, shape, CL, etc., which would not be feasible?), so less energy gets thrown away in downwash.(But aren't longer blades better for induced loss, if all else is equal?) But all other things are not equal. Some propeller designs are more efficient than others. If your blades have a round tip or a wide tip is it less efficient than a blade that has a very narrow or zero chord tip. Do your blades have a clunky, non-aerodynamic shape in the root just outside of the spinner, with a flat, triangular area about 2"-3" long top and bottom? Then it is not as efficient as a blade that has the correct airfoil shape at the correct helix angle all the way to the spinner. Does your spinner have wide clearance around the blade or is it sealed up to the blade? Big holes make drag! All of these factors enter into a prop's efficiency, so comparing two blades of different diameters, planforms, and streamlining is futile!
This extends also to the back-and-forth about CS vs FP. Even though you can operate the engine more efficiently at high MAP and low rpm with a CS, if the CS blade has any of the drawbacks just listed, a well-designed FP can actually give more efficient cruise at a higher speed!