For a long time, I've been a one person startup, and have just bundled all business costs - presenting at conferences/travel, office rent, hardware equipment purchases, etc., to my personal account. Currently, I'm in some considerable open credit card debt due to that. How complicated would it be if I were to transfer the debt to a business?
You can't transfer the debt from you to your (presumably newly created) business because you would need the consent of the company who owns the debt, and you aren't going to get that kind of consent.
If your purpose is reducing tax burden, then there is no difference between owing the debt directly and having a single-member LLC owe the debt. The best thing would be to get some income from this business so you can reduce your tax burden with the debt as an expense.
If your purpose is to file for bankruptcy to eliminate the debt, then you can't do that. Otherwise, all debtors would do it!