The only way the canonical link element would help is if the pages can still be indexed at the old URL - the one that uses the brand name. If you move the site, that won't happen.
There are lots of threads here about moving an existing site to a new domain. For a basic primer, see Moving to a New Domain - Official Advice from Google [webmasterworld.com]
Done with meticulous attention to technical detail it can be done with only some disruption to traffic - for a minimal period of a few weeks, maybe a month or two. A key to any such move is 301 redirects, URL by URL, so the business needs to maintain control of the legacy domain name for about 180 days, according to Google.
If the business cannot legally maintain control of the domain name, then there's a much bigger job and much longer traffic disruption. Two essential steps: publicize the domain change in a major way, and contact the backlinking website and gett them to change the links.