I take it your Name is Trademarked? You’ll need this before you apply for Brand Registry.
First things first, is to manage your inventory. Ideally, you really want to run stock down of the old before bringing in the new, but this is sometimes harder to manage in practice. Obviously you never want to go out of stock so there will definitely be a period of old and new stock but your aim is to minimize this as much as possible.
When you have new stock cleared at Amazon, that is when you’ll make the changes to your listing and SC name and email series and you may like to follow this guidance:
To change your Brand Name in Seller Central – All you have to do to update the brand is to change the “Brand” name on each product. If you want to show the brand name as the new brand and have the brand registry on ALL products (even if they aren’t updated yet), it probably wouldn’t cause any issues, because most people don’t look at the brand name. Just so long as you don’t change the information on the listing before the new stock arrives.
Listing changes – You want to let customers know that there is a brand name change, especially as the packaging might change from the old and the new brand. Remember this is an exciting and positive change and a lot of MAJOR Consumer Brand change their names all the time and they are positive about it. So on your listing, amend a bullet to communicate either ‘Brand X formally know as XX’ / ‘exciting rebranding’ other something similar
Email/Communication changes – Add a line of copy in each of our emails in your email series, with something along the lines of … “We are excited to let you know that X brand was recently rebranded to XX because it’s a name we feel connects better with our customers. It will always be the same great product” or something like that.