The problem (ignoring quantum mechanics) is that the sensors all require an EM field to operate in. So assuming that the speed of light was weighted with a vector in space-time, it would be affected everywhere -- including in the measurement apparatus.

If on the other hand one could detect a photon by sending out a different field, maybe a gravitational wave instead... well it might work, but the gravitational wave might be affected in exactly the same way that the EM field is affected.