We have a similar situation, but instead of merging two teams, a team in our area has lost their lea mentor and the team will now move in with us. IDK about changing everything, since we just started to make a brand of ourselves, but you could probably just ask them what they want to accommodate them. Maybe they don't want to have their name and logo on everything, or they wouldn't want everything shared. On the other hand, maybe they want everything changed. I hope not in our situation.

So, bottom line, ask THEM what they want.