A while back I asked this question Freely falling frame and the use of special relativity? which I later answered using the Hypothesis of Locality which I used to state that $E=-p_\mu u^\mu$ in any reference frame. My concern now is that the measurement of $p_\mu$ cannot really be done at a point and relies on at least two separate space time events. So does my argument in the answer to the linked question stand, given this non-local measurement - either way please can you explain.
I guess I am asking: Can we use all the laws of special relativity at the point of the 'observer' in all GR reference frames - independent of how many derivatives they contain?