You cannot claim a deduction without money "exchanging". So, in your case you could charge and collect the FMV rent and then make a charitable contribution to get back to the target rent.
It is similar logic to reducing rent for the tenant providing a service or paying for a repair. You have to claim the full rent amount then take the deduction as an itemized expense. If you are paying for a service provided by the tenant, then the tenant is obligated to report it as income (and for you to deduct $600 or more you would need to file a 1099).
