From my own experiences, and the various answers available on Amazon forums, Brand Gating is a black box. There is no one you can talk to in order to request it, and for now it seems to be an invite-only program.
Somewhat perversely, someone claimed on the forums (I’ll try to add the link if I find the topic), that one way to get brand gated is to buy up the counterfeit products being sold under your listing/ASIN, and document and/or send them into Amazon. It seems that until you’re being widely counterfeited, there really is nothing you can do proactively.
Amazon has been an overall net positive for our company, but the difficulties of trying to get a knowledgeable person to answer pressing questions usually create headaches for me at least once a month. Even as a large volume seller through Amazon Vendor Central, we’ve been given boilerplate legalese about needing to provide photocopies of the certificate of our trademarks in order to initiate DMCA takedowns of stolen or copied images and text. As great as Amazon is for consumers, the vendor support can often make you feel like you’re howling at the wind.