Re: "...using methods that can be replicated anywhere on the planet."
Not so. You simply define your local second, the second in your frame of reference, and adjust it appropriately for other frames of reference (inertial/gravitational).
As long as your cesium atom and your experimental equipment are in the same frame of reference, you'll get the same value for the second as anybody else in the known Universe.
There is a slight uncertainty due to the variation in gravitational field strength at different parts of the apparatus, but this in fact makes your measurement more precise and reliable if your experiment is in space, where the gravitational field gradient is minimal.