Excellent question. There is still some - maybe just a little, now - SEO advantage for having a keyword rich domain name. With that said, don't let it be a show stopper./td>
You might suggest that both domains are purchased. When you set up the hosting account, point both domains to the new host and set up a domain alias for the branded URL (Frankies-Fine-Cuts.com).
Set up the SEO friendly domain (hairdressers-hometown.com) as the main website. Then, simply point the Frankies-Fine-Cuts.com to hairdressers-hometown.com with permanent 301 redirect.
That should do it!