It is hard to know what you are dealing with without knowing where you are talking about. I am assuming that if they are checking backgrounds you are either in a condominium or a strict HOA like Harbour Island. If the tenant is unworkable - try taking just a small discount on your monthly rent and get another renter quick to ease the financial burden. Good-luck.
Read the guidelines that the HOA are using to determine the qualiifications of the applicant. Talk to the Board of Directors or Management Company about this matter. Exceptions are commonplace and if you try, you will succeed
If you let them move in without approval you could be subject to fines, a lien placed against the property and other legal actions.
If they have grounds to deny the tenant you should get that it writing and check to make sure they are not violating law.
Feel free to contact me directly if you have additional questions. tony@TampaNativeAgents.com