I didn't notice that the forums doesn't like Tor. Currently, it's just not really a thing for Tiny Core I'd guess. Not saying it can't be (!) - Excluding certain Firefox files from the backup, plus using some extra privacy plugins and tor proxies (or the browser provided by Tor/Onion, if any) might probably give you a similar result to what TAILS provides, but in that case one could simply use TAILS, where decisions where made strictly with security/anonymity in mind.
To be honest though, for me, one of the most important aspects of Core is the fact that I can reboot and my Web Browser is thinking it was freshly installed, except for my personal "default" settings that got restored via backup.
Actually, "Tiny Tor" might be a great respin for the extra paranoid.