But the thing is, Charlemagne really took nothing. Sure, he took over control over most of the barbarian lands in the west, he did do that. But he didn't "take" the title of Roman Emperor. He would have had to go to Constantinople to do so.
Yes, he did take the title. He and his descendants used it until the early 19th C. The fact that the Eastern Empire may not have recognized it was not going to lose them any sleep.
Look the question was creating a pentarchy-like decision in the west. I objected because the people involved would (many of them) side with the HRE as already possessing the right to the title. The fact that you do not believe they had such a right is really irrelevant, just as irrelevant as my own belief that the whole line of Roman emperors was illegitimate. The question is whether this idea makes sense in the context of time and place. I don't believe it does.
I understand that you care deeply about this subject, and I'm all for people arguing for their beliefs. (Further, I have a good deal of sympathy for your point of view.) But always remember that, in historical arguments, it is strictly impossible to prove oneself certainly correct, as one can in geometry. It is interesting to see what the Byzantine view of the HRE is. It can be supported by argument, of course, but it is always subject to qualification and refutation. That's inherent in the nature of history.
Again, I sympathize with your position, I tend to argue the same way when the Stuarts are brought up. But note that it does no good to bring up things which weaken your case. 2 examples:
1. Anyone who thinks Runciman is biased against the Byzantines is showing his own, stronger, bias in that direction. It weakens your case to imply this.
2. Earlier you referred to the Roman Catholic Church as being "under Orthodoxy" before the schism. This too is really tendentious. Before the schism, they were united in one communion. It is my opinion that both Leo IX and Michael Cerularius come out badly, but that is not a fact.
But my points are not to dispute your position, per se, but to insist that (a) most of your argument is, in fact, irrelevant to the topic of this thread, and (b) to warn against building into your arguement points which are, or were at the time, themselves in dispute. I repeat, I have much sympathy for your views, but you need to tighten up some of your arguments.