Equally a problem with the fine structure constant, since it directly measures the strength of the electromagnetic force. However, while the length of a platinum bar would, as you say, change under a change in the speed of light, it would not change linearly; so you can measure ratios of lengths and presumably pinpoint the constant that changed. The curvature of the Earth, for example, presumably depends on the strength of gravity in addition to the electromagnetic repulsion of its constituent particles, so it should change more slowly than the platinum bar's length.
Equally a problem with the fine structure constant, since it directly measures the strength of the electromagnetic force.
But the fine structure constant is dimensionless, so you don't need to find a standard to measure it which doesn't change when it changes.
New Salt Compounds Challenge the Foundation of Chemistry
The title is overblown (it depends on what you think the foundation is), but get a load of this:
And here's the philosophical bit:
The obvious example of local truth is relativistic effects being pretty much invisible over the durations and distances that are normal for people, but there's also that the surface of the earth is near enough to flat for many human purposes.
Any suggestions for other truths which could turn out to be local?