Re: "...using methods that can be replicated anywhere on the planet."
There are measurements that suggest those constants are in fact not. Constant, that is (cannot remember which ones, though)
There are some observations that suggest that Planck's constant ("h") might not be, er, constant in space and time. Time is sort of understandable, at least in the very early phases of the universe during expansion but I think the observations are of much more recent times. Came across this a few years ago so it could have been debunked or theorised away but it would not suprise me in the least if the universe didn't have quite a few more surprises for us.