I wanted to use Chrome but it looks like Chrome for Android can't run extensions? And support in Firefox seems shaky.
All I want is to request and execute remote CSS and JS files for a particular domain. I don't need any UI or anything.
Not sure if I should just forge ahead with Firefox or perhaps think about rolling my own 'browser' with plugin embedded?
You can use the Firefox SDK, available here: https://developer.mozilla.org/en-US/Add-ons/SDK
It allows you to develop for both platforms. And if you have troubles with sdk in firefox for android, you can read this: https://developer.mozilla.org/en-US/Firefox_for_Android
If no UI is required, it will be easy to set up :)
Regarding comments in this post, I made a script for setting up the under development extension: