So, what’s the reflexive equivalent of non-binary (specific) or gender neutral (non-specific) they?
They are different, btw: you use the former for someone non-binary, and the latter for generic non-gender referents. E.g. Sam (who is intersex) recused XXX from…, vs. The chairperson shall recuse XXX from… The former is much newer than the latter.
Two clear solutions: James Cottam’s theirself, or OP’s tentative themselves.
James’ solution has the advantage of being accurate as to number. It has the disadvantage, (a) of being a novelty (which is precisely why they has proven so popular—it isn’t), and (b) disagreeing with the number of they that you are using everywhere else anyway.
Sam said that they were happy to observe the impact on their commitee [plural plural plural, familiar pronoun], but they [plural] have [plural] to recuse theirself [singular, and unfamiliar new pronoun] from any vote.
Myself, I’d go with your first instinct, OP. Themselves. You’ve assigned the plural as the subject already, that battle’s lost in all other ungendered references.