Let's flip the question and ask what happens if we let the value of c depend on the location in spacetime.
Ellis provides a "short checklist of issues that should be satisfactorily handled by" theories that have a value of c that depends on location in spacetime. https://arxiv.org/abs/astro-ph/0703751
The tl;dr version is that relativistic physical theories will need revising because in the face of a location-dependent c, and in particular the problem arises because we necessarily break lengths and durations when c is not everywhere-and-everywhen identical.
More narrowly observables like spectral lines are sensitive to ratios involving the quantity hc which are taken to be constant. Since cosmological redshifting of spectral lines fits in a substantial web of observables related to distance (e.g. angular diameter, surface brightness) we can practically rule out variation in hc to a very high redshift (z > 10).
You might instead be asking what happens if c is the same everywhere but we've settled on a very slightly wrong value of c. The answer is that this will almost be absorbed into our system of units, in particular in various ratios involving hc. If we change the value of c to the slightly improved value, we also probably slightly change things like the fine structure constant and the electron-to-proton mass ratio, which are dimensionless quantities with ratios involving hc. These dimensionless quantities are good checks on the many ways in which we might measure c empirically.
Ellis provides a "short checklist of issues that should be satisfactorily handled by" theories that have a value of c that depends on location in spacetime. https://arxiv.org/abs/astro-ph/0703751
The tl;dr version is that relativistic physical theories will need revising because in the face of a location-dependent c, and in particular the problem arises because we necessarily break lengths and durations when c is not everywhere-and-everywhen identical.
More narrowly observables like spectral lines are sensitive to ratios involving the quantity hc which are taken to be constant. Since cosmological redshifting of spectral lines fits in a substantial web of observables related to distance (e.g. angular diameter, surface brightness) we can practically rule out variation in hc to a very high redshift (z > 10).
You might instead be asking what happens if c is the same everywhere but we've settled on a very slightly wrong value of c. The answer is that this will almost be absorbed into our system of units, in particular in various ratios involving hc. If we change the value of c to the slightly improved value, we also probably slightly change things like the fine structure constant and the electron-to-proton mass ratio, which are dimensionless quantities with ratios involving hc. These dimensionless quantities are good checks on the many ways in which we might measure c empirically.