First off, let me say that your question very likely needs a lawyer and I am NOT a lawyer (nor am I trying to give you legal advice). But, I do have 10-years of leasing experience across all product types from commercial to residential to retail.
In my experience clauses like this need to be pretty explicit in how and when things occur. I personally would like to see something that says the tenant will "cooperate". I also don't like any clauses that say if a deadline passes the other person gets to do something (i.e. reduce rent) -- you never know if they want the AC unit or just a discount on rent.
With that said, I don't know what your lease says or what options you have but I would think you should determine whether the rent discount is worth the savings on the AC unit. Maybe just take the hit on rent and save the cost of a new unit?
My leases always say that the tenant can't change anything without my permission and/or supervision. So, assume yours says something similar, I'd also let them know that the tenant will need permission for any work done. If they do rebate rent on their own, I'd give them a short-payment notice to record officially that they underpaid (in the absence of a receipt or proof of payment, you have nothing to verify they did the work).
It sounds like a difficult tenant. Maybe just give them an option to get out of the lease?