Tattoos that signal group membership are a costly signal of loyalty, and the cost is precisely closing those other doors. It's just, with the "rationality tattoos", I am not sure what exactly one gains in exchange for paying the cost.

I'd say that a truly rational tattoo is the one that can be easily removed. :D

I could imagine a specific situation where having a rationality tattoo could be the rational thing to do (for reasons other than impressing people who are easily impressed by tattoos), but those are quite unlikely situations. Having a rationality tattoo doesn't reliably signal rationality -- even a stupid person may decide to get one -- so what exactly would be the purpose? Signalling hostility towards groups that openly identify as anti-rationality? Not sure there are many such groups.