Measuring components of current
Measuring the load and no-load input power of a transformer gives you a reasonable idea load power, but it's not strictly accurate.  The losses in the transformer are not constant and will depend on what the load does, even with a fixed input voltage.

The best way to measure load power is to measure load power.  Put the RMS meter on the secondary of the transformer to measure it directly.  If you are trying to measure losses in the transformer, then you want to measure both input and output power.  The output power will be less, and the difference will be the losses in the transformer.

Another less invasive but more elaborate way to measure the losses would be with calorimetry.  Just about all the power lost will turn into heat.  If you measure this heat power, you will know quite accurately how much the transformer is losing.  If you also measure input power, then you can calculate output power.  Note however that the calorimetry method of measuring loss power is inherently very low pass filtered.  Put another way, it will take seconds to minutes for the reading to catch up to a new steady state.