Do you know if pacman is available on Chrome OS? My understanding is that it's based on Gentoo but it's heavily modified. So adding a different binary-based package manager isn't that crazy of an idea, especially for Google.
Two package managers are a bad idea. They would need to cooperate somehow. Pacman developers probably have heard of Gentoo, but I wouldn't call it "based on Gentoo".
I meant that since pacman is not available, then use a different one entirely (not both). It seems it's a moot point b/c of the sibling comment saying the disk is read-only.