The answers are "not really, and no".
Special relativity is based on the two postulates that (1) the laws of physics are the same in all inertial frames of reference, and (2) that the speed of light in vacuum has the same value $c$ in all inertial frames of reference.
The reason we express $c$ as $\approx 300,000$ km/s is that we chose the units kilometre and second, but we could just as well use units of length and time that make $c=1$. So you cannot just change $c$ on its own, since it would just rescale distance and time. Normally when physicists consider different values of the fundamental constants they look at dimensionless constants, that is values that do not have any units and hence cannot be rescaled by changing your system of units. So changing $c$ requires changing a few of the other constants, changing a fair bit of physics. However, let's ignore that part to get to the question - just assume we mess with the constants in the right way.
Does the fact that light travels at $c$ do any work here? Apparently not: the only thing needed is that there is some signalling speed that is invariant in inertial reference frames. The causality goes the other way around: since photons are mass-less (or, alternatively, Maxwell's equations are Lorenz-invariant) light have to travel at the invariant speed. Relativity would still hold if space had a refractive index $n>1$ slowing light down.
The first question is whether in a universe with a different value of the invariant speed $E=mc^2$ would have to change. The quick answer is no: in this universe $E=mc'^2$ where $c'$ is the changed speed (if you want to express this formula in terms of normal $c$ you will need to add a factor in front of it, but now you are expressing one the observed constant in terms of one from another universe - it doesn't make much sense, and you will have to do it everywhere $c'$ shows up in your equations).
The reason is that the derivation (variants are found in all relativity textbooks) only makes use of how momentum and mass transforms based on the invariant speed, not what value it has. It can just be treated as a symbol: there is no link to actual light or a particular value.
(Strictly speaking, Einstein's original derivation was a lot about emitting photons and seems to make light much more important to the result than it is.)
The second question is whether we could end up with an equation like $E=2mc^2$. The answer is no for the same reason. The math will not work out. In particular, consider the energy-momentum formula $E^2=(mc^2)^2+(pc)^2$ - if you want this to hold and $E=2mc^2$ then you get an imaginary momentum. So unless you want to postulate a universe with a really different physics you are stuck with $E=mc^2$.