Rebranding: How to build a new brand without disturbing existing customer relationships...

I'd like to pick your brains a little on the topic of rebranding.

Our company is a (large sized) small business with a few locations around the PNW. We are being purchased by a very similar company that operates in a different region of the country. While our company will function very closely to how it does now - it is highly likely that we will change the name of our business to fall more closely under the umbrella of the parent company - while still maintaining a bit of our own identity.

Changing our name (and subsequently our branding) makes good sense for a few of our branches that have struggled to gain momentum in their local market place, as they will now be more healthily supported by this new parent company. However, our main branch location has a newer business segment (launched approx 3 years ago) that has worked very hard to develop a 'name' for themselves in their segment of the market place, and they fear that changing our name will diminish everything they have worked hard to create as far as brand recognition for their segment.

I feel as though a fresh start would be great for our company as a whole and positioning this merger as a huge benefit to our potential and existing client base will be a good marketing path for this transition. However, I'm not 100% sure how to proceed with the one business segment that feels they have developed a lot of valuable brand recognition for themselves that will be completely lost if we rebrand.

What advice do you have that could help ease this transition for our business, so that we don't abruptly lose the current recognition our company has but rather continue to build on that foundation with our new brand?