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One of the (two) core assumptions that Einstein made with special relativity is absolutely that the speed of light in vacuum ("c"), is constant in every inertial reference frame/coordinate system. This has been borne out experimentally and the consequences of special relativity are seen every day at any particle accelerator (for e.g. time dilation of lifetimes of particles).

The full equation is:

E = mc^2 / sqrt(1 - v^2 / c^2)

This is coordinate/frame-dependent since it depends on speed. Here, m = mass of the particle which is also the same in every reference frame. You could define an effective mass as m' = m / sqrt(1 - v^2 / c^2) but that obscures the point imo.

To address your point directly, when you say "as the energy to mass ratio goes up, the speed of light goes up", that is just not true both mathematically and physically.