The problem with having a Dr. Chien for insurance questions is the law is different from state to state.
Subrogation is in general terms (again, this may vary, depending on strate law) a legal remedy usually afforded both under the common law (meaning established by legal precedent issued in written appeals court opinions) and expressly as a term in the applicable insurance policy. When the insurance company pays its own insured for a loss, it then may attempt to seek recovery against the responsible party. Thus, when the FBO's insurance company pays the FBO, usually nothing is stopping it from sueing the responsible party, namely you, the pilot operator. But, when they do so, the "stand in the shoes" of their subrogee. This means that any defenses you would have as to the FBO, you would have as to the carrier. And some scenarios (construction projects, for example) parties will include a waiver of subrogation provision in their contract which effectively binds their carriers, and prevents the carriers from sueing the other party. So, if you can manage to sneek a waiver of subrogation provision into your rental agreement, you may at least get that benefit. Although that won't afford you liability coverage for the damage you cause on the ground.
In some states, mine for instance, the common law holds that an agreement to provide insurance reflects an intent of the parties to place the risk of loss on to insurance, and that therefore the parties agreed to a waiver of subrogation. Thus, a representation by one party in the rental agreement that they will obtain insurance for a certain risk may be enough to argue that the parties agreed the loss would be covered by insurance, and not the parties to the rental agreement, thereby depriving the carrier of the right to subrogate.
