what happens to the electron cloud at very high atomic number, when the innermost electron would, using a non-relativistic model, be calculated to exceed the speed of light

I can only tell you that when a atomic level is hig it as more electro around the electron cloud and it means that it is harder to be unstable

the old alchemists made significant contributions to science even though there starting premises were off a bit. their conclusions after failure would steer them into better directions for investigation. science is no more than close observation and the initial mistakes result in rethinking and retrying experiments.