A first-offense DUI isn’t a felony in most states, IIRC.
I recall a story from a few years ago when then Sen. Robert C. Byrd was in a traffic accident while driving through Maryland, unrelated to Congressional business. He was at fault and cited for some petty traffic violation, but the police department then withdrew the citation because of the Constitutional clause.
IIRC, Byrd was embarrassed by the special treatment and attempted to waive his protection and accept the citation, but the thought at the time was that the privilege could NOT be waived. Maybe my google-fu will turn up something.