Perhaps, any awkwardness you might feel towards “célibataire” comes from “celibate” being related to religious beliefs in English? As far as I know, there is no stigma or negative connotation attached to “célibataire”.

I have used “Êtes-vous célibataire ?” a couple of times myself in the not-really-trying-to-ask-them-out-but-rather-out-of-curiosity context, and without any problem.