Vaan wanted to get invited into the Marquis house. By telling everyone he was the long dead captain Bash (that is what the Marquis told everyone) he earned himself an invitation.

Basch, as you noticed, wasn't really dead and the lees people knew it, the better for the Marquis. Hence he invited Vaan to tell him to STFU. And then the Marquis delivered Vaan and his friends to the Empire (the Leviathan was an imperial ship taking them to Archades to rot in jail). The Marquis is not that happy with the Empire but can't afford to oppose them openly.