No no no a million times no. There should be one and only one point of contact for booking with a particular venue, otherwise you are going to confuse people and you are going to cross lines of communication and get stuff messed up.Squid pro bro: when you see him give a card you give yourself as well. You can both get to know the contacts. Just have a discussion able how to best present yourselves.
You do need to address this behavior with your bandmate. It's childish, seems ego-based, and is already detrimental to the band and has the potential to become worse.