Signatures do nothing to prevent blind relaying. Transmitting through an amplifying relay in this attack looks identical to transmitting through free space, aside from the received power level and propagation delay.
The amplifier relays that signed signal to the fob. The fob receives it and its unlock signal is relayed back to the car.
No matter what tricky message protocol you come up with, it won't matter. The car and the fob can't detect the difference between being next to each other, and being next to a set of relays rebroadcasting their signals. Not by reading and transmitting radio signals at least.