it would have essentially become one band with 2 different shows.
I played in a band that had an alter-ego: Same four guys, and sometimes, the same material. The only difference was, if we'd had any beer to drink before going onstage, we referred to ourselves by the alternative name. If we went on sober, we used our normal name.
On those rare instances when a promoter had actual done any advertising with one band's name but the other band showed up, we'd just announce "Sorry, ________ couldn't make it so we're filling in for them."