There is currently no profit being generated on the property at this time.
That's normal for long term residential rental property. It is not common for long term residential rental property to ever show a profit in any year "ON PAPER" at tax filing time. Especially if there's a mortgage on the property.
Is there a tax benefit to adding the property under an LLC and claiming business losses rather than rental property losses?
No benefit at all. An LLC is nothing more than a pass-through entity. Putting the property into an LLC changes nothing on the tax front. The rental income/expenses still ends up reported on SCH E on your personal 1040 tax return.
