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I should probably expand on my earlier response a bit...
Point 1. It's just as important to describe what you aren't changing as what you are. For instance, c = 1 / sqrt\left(\epsilon_0 \, \mu_0 \right). So, obviously if you're changing 'c', you can't keep both these fundamental constants the same, and you need to specify if you're changing only one, or if you're changing both, how they change. It's also not entirely clear what other physical constants you may or may not be holding constant when you "vary c", the procedure operationally isn't well defined.
Point 2. You might be making a simple "scale change", without even realizing it. For instance, suppose your change of 'c' changes the size of every atom in the Earth uniformly. Maybe what looks like a change will vanish if you define a "new meter" that has the same number of atoms laid end-end as the old meter. This is assuming you are using the old-fashioned meter stick standard rather than the SI definition, which you probably are, because if you were using the modern definition you wouldn't be talking about varying it, because the new definition just assigns 'c' a constant value.
Point 1. It's just as important to describe what you aren't changing as what you are. For instance, c = 1 / sqrt\left(\epsilon_0 \, \mu_0 \right). So, obviously if you're changing 'c', you can't keep both these fundamental constants the same, and you need to specify if you're changing only one, or if you're changing both, how they change. It's also not entirely clear what other physical constants you may or may not be holding constant when you "vary c", the procedure operationally isn't well defined.
Point 2. You might be making a simple "scale change", without even realizing it. For instance, suppose your change of 'c' changes the size of every atom in the Earth uniformly. Maybe what looks like a change will vanish if you define a "new meter" that has the same number of atoms laid end-end as the old meter. This is assuming you are using the old-fashioned meter stick standard rather than the SI definition, which you probably are, because if you were using the modern definition you wouldn't be talking about varying it, because the new definition just assigns 'c' a constant value.