Having played with both, I can probably have a go.
Mirror lenses have some things going for them - they are inexpensive, compact, light. On the other hand they tend to have poor-to-indifferent optics, they are optically slow and have fixed apertures. The bokeh - particularly on highlights - looks like a donut. Some like it, some do not.
I have a Celestron 500mm f/5.6, a 1980's derivative of the C90 spotting scope. On the whole it's better than not having a 500mm lens, but it suffers from chromatic aberration. Provided I get the focus right, I can usually fix this in post processing.
A slow, fixed aperture lens means that you may not have enough light in the viewfinder to focus easily.
On the 70-200/2.8 + 2x teleconverter.
You will find that this combination is a long way from either the 400/5.6 prime or the 300/4L + 1.4x tc. I've tried all combinations and found that the 400/5.6 was sharp and quick to focus (but it was impossible for me to use because of camera shake.) The 300/4L + 1.4x tc worked well - the IS makes the combination less user hostile than the 400/5.6L. Finally, it seems that almost every lens suffers when a 2x converter is attached.
Last thought - you could also consider the 100-400 - it's a decent lens indeed.