I stumbled upon this post which explains how to burn the ISO of a different Linux using ChromeOS's Chromebook Recovery Utility: https://chromeready.com/3742/burn-iso-u ... hromebook/. Basically it say to rename the ISO to bin, e.g. fossapup64-9.5.iso to fossapup64-9.5.bin. Then run the Recovery Utility and select that 'bin' file where the Utility offers to "Use Local Image".
I don't have a Chromebook to try this on. And I suspect that the resulting Puppy will not be able to write SaveFiles/Folders on the USB-Key. But, if it boots at all, you should be able to use that Puppy, it's builtin gparted, and frugalpup, viewtopic.php?p=1588#p1588 to produce a fully functioning USB-Puppy for PCs. Will it run on a Chromebook? Haven't the foggiest.