I looked at synchronous sampling in detail once. I was going to sample at 240 kHz as the lowest frequency to give quadrature data (slower is lower power). The input stream is multiplied by a pair of sine waves to give the I and Q streams which at 4x the carrier rate are just sequences of 1, 0, -1, 0. After applying a simple filter the I,Q samples become (y(t)-y(t-2), y(t-1)-y(t-3)), so no math other than subtraction.

There are short cut ways of taking the RMS of the samples (amplitude signal) and dividing to get the tangent. I can't believe I didn't realize 180° phase shift doesn't change this ratio. Hmmm... someone double check me on that. Yeah, my calculator says they are the same. So the sign has to be checked and that means both the I and the Q have to be checked since one could be very close to zero.

While the amplitude squared can be accumulated as a running sum, the I and Q parts probably should be accumulated separately then the signs and phase angle calculated at the end of a sample period. You want to track the phase angle (the tangent) to see how your oscillator is drifting wrt the carrier. I was planning to integrate over 0.1 seconds. The AM varies with a resolution of 0.2 sec, but the phase is constant for a full bit time, 1 second. The phase changes when the amplitude is low so aligning to the edge of a second may be hard other than by using a long term integration. Such a long term integration is not amenable to the lowest power levels, but is fine for wall powered equipment.

Still, if you are trying to get the most accurate timing possible and you are not close enough to the transmitter to receive a good signal when the amplitude is low you may be limited in the accuracy you can achieve.