Surely the answer then, is to use a gravity meter and park it alongside the chronograph, then you will be able to use the actual g appropriate to your locality rather than rely on a nominal figure for g?
Nope, the answer is to use a properly defined unit of energy e.g. Joules.
Some of the problem cancels out if you weigh the bullet using balance beam scales, then local g doesn't change the reading i.e. you get a reading of mass, not weight. Beam scales will give the same reading in any gravity field.
But use digital (strain sensor) or spring scales. Then the weight you measure depends on the local value of g. So you truly measure weight, not mass.
Taken to extremes, shoot any bullet in a zero g environment and it will have exactly zero fpe muzzle energy, as measured in that gravity field.
fpe is an archaic measure that is only meaningful if everyone agrees to use an arbitrary figure for g. Its not a proper measure of anything, but its all that we have (until we adopt e.g. Joules).
That would be fine, but it seems few agree on the value, so we have 450200, 450240, 450230, 450400, 450430, etc. etc. all used.
However in small arms ballistics the old imperial measurements are very well suited to the practical problems. Measurements in grains, 'thou, feet, inches at 100 yards, MOA, fps etc. just work nicely. I'd hate to have to convert to grams, millimetres, metres/sec etc, the units are just wrong .