Not sure about this particular car, but the idea that a roadster receiver is better than a coupe's is interesting. You would think that the coupe transceiver would be designed to be stronger b/c of the metal shell vs fabric roof on the roadster.... Maybe they are somehow using the roof to augment the antenna...?
One small note if you do decide to increase the transmitter's antenna length. These are usually precisely calculated traces on a pcb. (The tuning circuit usually uses the impedance of the antenna along with a small tuning capacitor to set a transmit frequency [gross oversimplification but basic idea]). If you were to just splice a random piece of wire, you may actually decrease your range b/c you may increase the amount energy reflected back...
If I recall correctly, the proper way would be to figure out the exact length of the transmitter and add a length of wire that is a multiple of that (eg: if antenna is 2m now, add another 2 to make the total length 4m). Or if you know what the frequency of interest is, adjust the tuning capacitance to compensate for the change in impedance.
I recommend that you try playing with the receiver in the car instead...may be simpler!
If you want more info, try googling antenna length tuning and SWR (Standing wave ratio).
gas-can