Could it be something like
CYCLE? (I also thought of PERIOD in the same vein).
Dethrone the king, cast out the poorest
Revolution repeated summons me soonest
If the first line is a red herring, and "revolution" is meant in the sense of around a circle, then repeated revolutions could definitely lead to a cycle.
Put the rocks in the sky and the sun in the sea
Either way 'round, I'll still be me
The daily cycle, sun appearing to enter the sea at sunset? (depending on where you are) Not sure about "rocks in the sky" though.
I am (mostly) in crabs, and fish, and in birds
This way I am myself in these very words
Many birds and fish have migration cycles, and so do some crabs. There is some repetitive language (cycle-like?) in this very verse.
Fastest in each of a metronome's flicks
And part of its tastiest ticks
A metronome is of course flicking and ticking periodically on a cycle.