Enter your business address for the Payee and mail the 1099-MISC to the Payee in a manually addressed envelope. The IRS doesn't much care about the address on the 1099-MISC - they just want the SSN to be correct.
(Actually, the only reason the Payee's Address is on the Form 1099-MISC is for your mailing purposes).
The recipient will file his or her tax return and provide the only address the IRS needs - the one on the return.