That's why you build another instrument that is demonstrably a factor of two more sensitive -- it either detects the new thing with much higher precision or credibly disproves the old measurement and makes forward progress at the same time.

If you can't do that, then you build another instrument that uses a different technique but reaches similar sensitivity. If the two instruments disagree, this will tend to lead to confusion and general wringing of hands. If at all possible, do the aforementioned higher-sensitivity plan.

"That's why you build another instrument that is demonstrably a factor of two more sensitive"

This is not possible

It definitely is. In this specific case, LHC will almost certainly yield improved mass measurements. Just as Fermilab published this result a decade after the Tevatron shut down, increasingly refined measurements like this will continue to emerge from LHC through ~2050 or so.