A decent frequency counter can resolve a frequency difference of 1E-15 in a few minutes of averaging. Modern GPS-based frequency comparisons (dual-band, post-processed using precise point positioning) are also accurate to the 1E-15 level with around one day of averaging. Note that these counters operate with radio frequencies. The best atomic clocks use optical transitions and are accurate to around 1E-18. In this case, one gains around 6 orders of magnitude in frequency resolution by translating the laser frequency into a radio frequency using an optical frequency comb. The accuracy of the relative frequency measurement is not limiting at all.

The gravitational redshift amounts to around 1E-16 (not 1E-15) when moving the earth one meter closer or further from your clock. You standing next to it is going to have absolutely no effect.