Disrupting a funeral is one thing. Disrupting a funeral the public is forced to pay millions of pounds for, while other members of said public can't afford food and electricity, is a completely different matter.
I don't think that it is. The rich and powerful are still real people with real grief. I'm not opposed to people protesting parliment or whatever, but let the mourners mourn in peace.
Is it your view that just because the state pays for something the public has every right to do whatever they want with it?
This seems obviously false. Try walking into the petagon some time proclaiming "the public paid for this military base and therefor i have every right to participate in it." And see how far that gets you.
If this were some kind of private function I'd agree with you. But a large, public funeral for a monarch is inherently a political event. It's part of the process of moving from one monarch to the next, which is exactly what the protestors object to, and is literally politics. It is absolutely fair game to protest.