I would think it is a simple matter of a current probe and a voltage

probe, one on each channel and measuring the offset to calculate

the phase angle.

Is this a reasonable method given that your probes are properly

calibrated?

Yes. You'll probably need to take a picture of a steady waveform

so you can go off and measure it accurately. Measuring the offset

is only useful if both are sine waves. For many loads nowadays, the

current waveform can look very strange, and on many mains supplies

the voltage waveform is noticably flattened at the peaks, in which

case you will need to integrate by hand the product of the voltage

and current waveforms over a cycle to work it out.

OTOH, true power meters are the tool designed for the job, and

you can just read off the answer.