personally would have used a polariscope first then maybe the refractometer.
I then would have used a spectroscope to see if I could get a spectrum. even may have used this after the polariscope
I would think that with the RI and the Birefringence that you have said, if it were uniaxial negative that the gem would be apatite and I should get a good didymium spectrum.
No need to check specific gravity if that were the case as the spectrum would be diagnostic
please make comment if you wish thanks