This formula:
$$E = mc^2=\frac{m_o}{\sqrt{1-v^2/c^2}}c^2$$
means that the temperature of the calorimeter in an experiment Bertozzi continues to grow, while the rate remains almost unchanged.
I have info that monstrously huge energies that (as if) manage to get at accelerators exist only on paper. These huge energies are derived from this formula. In a calorimeter this monstrously huge energy generate a ridiculous zilch. It is this circumstance is the reason for the lack of results of these experiments in a public press.
I am confident that in an accelerator of an elementary particles get speed up faster the light speed does not allow the Doppler effect, but not the unlimited growth of energy and mass.
To me what is significant about this formula, $$E = mc^2=\frac{m_o}{\sqrt{1-v^2/c^2}}c^2$$, is that it is defining the $$m$$ in $$E = mc^2$$ as inertial or relativistic mass, rather than rest mass.
I have heard a number of prominent physists over the years explain $$E = mc^2$$. Some have explained it such that $$m$$ should be interpreted as relativistic mass and others as rest mass. Personally, I am partial to the rest mass interpretation, but I can understand the other side of the picture also.
The first or rest mass approach I believe is likely more consistent with how Einstein saw it, but that is really more my impression than anything else. The second or relativistic mass approach is consistent with special relativity, in that the formula for relativistic mass, incorporates the Lorentz Transformation. Neither, is contrary to relativity. They are just different ways of looking at things.
It seems that portion you are most opposed to, and I may be taking some liberty here, is that both versions include $$c^2$$. Doing so, both sets a limit to velocities, which cannot exceed the speed of light, and at the same time removes limits from energy, which can essentially become infinite in principle. (i.e. No matter how much energy you put in you cannot accelerate mass to the speed of light...) This is very similar to what your statement above is about,
...the temperature of the calorimeter in an experiment Bertozzi continues to grow, while the rate remains almost unchanged.
the energy continues to increase while the rate or velocity remains almost unchanged. That is what the formula says, or at least one interpretation, and it is completely consistent with both special and general relativity.
I don't see that either version is in conflict with relativity. And yet the question you keep asking, when you ask for experimental proof.., I am not sure there is a clear answer.
A few posts up AlexG posted a picture of an atomic bomb blast as an answer and yet though there is no argument that there is a great deal of energy released, we cannot measure it directly. We cannot measure even what happens at a much smaller scale in particle accellerators directly. Everything that occurs at both scales, we can only evaluate through the theories and models that we have. Those theories have been very successful. Will they be the same theories we rely on in a hundred years? Perhaps not. We seem currently to be running up against some difficult obstacles. But I am fairly certain that it is more likely that what the future holds will incorporate or be built upon what we know today, rather than completely overturn it.