The last option, the one Kukaku mentioned just as you were walking out the door, is that you could change your family name to Shiba. It would feel wrong at first, but... if your dad was a Shiba, then aren't you one anyway?
"We aren't banished." She said, touching her sheathed zanpakuto. "We're just... not around, and kind of disgraced. Coming in as the next big Shiba would be unusual, but we're an unusual family. People are kind of used to it!"
It's just a name, but it could be a useful name, a name with weight behind it. It's almost funny, how much of what you've done recently depends on names.