How can I get the velocity-power curve for a particular aircraft?
You are right, this curve has to be tested. Of course, during development a lot of simulation and wind tunnel testing is done to get the power required, but it still needs to be confirmed in flight test. The interaction of power plant and airframe can be quite complex, and only testing will ensure that the data is correct.

If it is not part of the manual, the plane's manufacturer will certainly have the power curves. Did you ask them for the data? And I would expect that you will find some data points in any handbook which will allow you to approximate the power curve. Start from the trim condition (basically, thrust equals drag) and calculate the curve with the help of a parabolic polar. Like that:

$$T = c_D\cdot q\cdot S$$

$$P = T\cdot v = (c_{D0} + \frac{c_L^2}{\pi\cdot AR\cdot\epsilon})\cdot v^3\cdot\frac{\rho}{2}\cdot S$$

Of course, now you need to get the zero-lift drag coefficient $c_{D0}$ somehow from one datapoint in the manual. Same goes for values like propeller efficiency or intake losses, because the listed power of the engine will not be what is effectively employed to generate thrust.

Nomenclature:
    $T$  thrust
    $v$  airspeed
    $q$  dynamic pressure
    $\rho$  air density
    $S$  wing area
    $c_L$  lift coefficient
    $AR$  aspect ratio of the wing
    $\epsilon$  the wing's Oswald factor