That would only mean we calculated c wrong

I mean, being pedantic a little, we don't actually know if c is constant, since measuring c is rather difficult. If c is not in fact constant in some medium or environment, then a huge number of things get very weird very fast.

So, yes, we could've measured c wrong. We just would have no idea if we did.

Source: Veritasium did a very fascinating video explaining this problem.