Is it possible to measure phase margin with the loop closed?
Maybe. What you have outlined is a Return Ratio and not Loop Gain. Usually close and mistakenly used interchangeably along with other approximations such as the 'open loop' method and variations using replica loading. Much of the confusion lies in the lack of proper definition of terms. Aol and Beta are terms with origins in ideal block modelling carried over onto the real world where they are not quite complete. Aol becomes a function of loading and of the feedforward element, here R2, as well as its internal impedance, ro. Beta is also dependant on loading as well as input impedance of the opamp... Another complication arises if the opamp is not strictly unilateral and an internal feedback path is active.

If you were to set V1=0 and in a simulation, drive Vfeedback (Vfbk) with a unity ac swept source and capture Vout, form Hfwd=Vout/Vfbk|v1=0, then drive the system backward with the source at vout and capture Vfbk, forming Hbkwd=Vfbk/Vout|v1=0, then LG==Hfwd*Hbkwd. See Ochoa et. al, 2019 Mid West Symposium on Ckts and Systems. This works in simulation since ac analysis in SPICE is linearized and the 'virtual short' at the amp input is not an issue.