Well, of course you should do business under your own business name. That's why you have a business name. I can't see any reason to use a different name, just because it happens to match the domain name of a site you are buying. Remember, what's important is the name that your customer sees. That's your brand .
Going further, I don't see why you are buying this site in the first place. I assume you plan to use it to promote your business in some way. Why would you use a site for that that's got a poor BBB rating? And why use a site to promote your business that was originally designed to promote some other business?
I'm sorry if I'm missing something here. But it seems to me that your best strategy would be to drop your plans to buy this site, and instead to focus on developing your own site: one that is designed specifically to promote your business, and whose domain name reflects that of your company.