I don't understand if you have a question or not.
In any case a "true" Antenna Gain Measurement is certainly one of the most difficult measurement.
Moreover, as per every measurement, the difficult increase as required accuracy rise.
After this apparently obviously preface, the way i've used is the following:
1) Answer this question: what is the accuracy of commercial (high cost) standard horns? A: +/-0.1dB (that is +/-2.3%) . Of course i cannot do better!
2) Perform a path setup with a reference antenna, a standard gain antenna, then record level on boresight. Then calculate Pathloss+RefAntenna Gain. All in dB !
3) Replace the standard antenna with the AUT, then record a directivity pattern on many cuts (at least 0°, 90°, 45°). Then perform the integration around the sphere, Then, from the itegrated, solid beam, calculate the Directivity Gain.
4) Now, knowing PathLoss+RefAntenna Gain + AUT Directivity Gain , and knowing the signal level, you may calculate the AUT efficiency.
Notes:
:arrow: The efficiency, also indicate with the greek letter ε, is the ratio Gain/DirectivityGain. 0<ε<1
:arrow: AUT=Antenna Under Test
:arrow: Of course , enviroment is reflection free, same wave polarizzation, signal is stable on both freq. and level etc. etc.
Don't hope to be very accurate... it's quite impossible.