Synchro with 1 phase grounded
A synchro is a transformer, there's no internal galvanic connection between the S and the R terminals, so neglecting stray capacitance, you can ground any one of the S terminals and any of one of the R terminals without altering the performance.
The usual way to use a synchro is to generate a three-phase waveform 120° between phases, use them to drive the three S terminals, and detect the angle of the rotor voltage against the angle of S1. The rotor voltage stays nominally at a constant amplitude.
The three S terminals are, in phasor notation, driven at the vertices of an equilateral triangle, which in normal use is symmetrical about the ground. However, if instead, we ground one of the S terminals, then it's clear that the remaining two terminals are at the two other vertices of the triangle, or 60° phase apart from each other. If you generate a stereo signal with two such waveforms, send them to S1 and S2 while grounding S3, your synchro will still be driven properly.
If driving only two phase like this, it's important to use zero output impedance drivers. With three-phase symmetrical drive, even if the drivers have a finite output impedance, as long as they are balanced, the drive will be balanced. If you ground S3, and drive S2 and S1 with drivers with an output impedance of Z, then your stator sees Z in series with the S31 and S32 windings, but 2Z in series with S21. This will make the rotation input to phase output non-linear.
If you must drive it the other way round, driving R and detecting two of the S with respect to the third, then you need two receive channels instead of one. Each channel will still have nominally cosine amplitude dependence on rotor position. I'd guess from reciprocity that they would be 60° phase shifted from each other, it would be simple to do the experiment to check that.
A common way to handle a synchro cheaply is to use an MCU with an internal capture counter. Generate an interrupt at 6x your desired excitation frequency, so 2.4 kHz for 400 Hz excitation, and run a 6-state Johnson counter from the interrupt to generate a steady 400 Hz three-phase waveform. They will be square waves, but that's fine. Lowpass filter the R output, and use the timing capture peripheral to measure the timing of the zero crossings, and hence the phase of R, with respect to one of the input phases. We use a single lowpass filter on the R output rather than three lowpass filters on the S drive inputs, as then we don't have to match the filters.