Your problem has nothing to do with it being a convertible or the size of the motor. It is an electrical issue. I am assuming it isn't a stock radio. I am also assuming someone hacked into the stock wiring rather than going to the ignition switch or battery to obtain power and did not provide a good ground path when they added a radio (stereo).
First issue is grounds. Your car has a lot of wires in it (just look under the dash), but it has half the number of wires it actually needs (or the number you would find in a plastic bodied Corvette). The reason is the ground the factory chose is the car's steel body. In addition to the steel shell your car left the factory with copper ground straps that connected electrically parts of the car that are electrically isolated, because of the rubber mounts for the body and the power train (I mention this because many shade tree mechanics leave them off when swapping motors).
In a plastic Corvette there are two wires for every circuit. Just like you would find in your house. The wires when laid out on the ground would form a loop when opened up. This is the basis for a simple electrical circuit. A wire goes to your radio from the power source (your car's battery) and returns back to the battery by way of the ground source (once again in your house it is the white wire, the black one is hot, and the green is neutral and has no juice in it; and should be isolated from your load).
The problem is two fold. First your car is old and someone before you may have cut up the wiring to install an aftermarket radio. If this is the case, you are on your own and no one here knows what he (the former owner) did, so it does no good to ask. Another issue with an old car is corrosion, when combined with wear and tear which breaks spot welds joining your car together; combined these two issues reduce the chance of a good electrical ground.
Rust (corrosion) is an oxide of iron and doesn't conduct electricity nearly as well as clean steel metal does (which is a poor conductor when compared to a copper wire, but it is a lot cheaper than running two wires; which is why the factory does it that way). If there is rust where the ground wire attaches to the steel of the car's body you will have a bad connection (such as where the lamp attaches to the body) and no return path.
Electricity is like water, it follows the course of least resistance. If it can not return to ground by way of the steel body it will find another path, or if the resistance is too high do nothing at all (this is what an on off switch does; breaks the circuit with a high resistance air gap). In your case you have power trying to go across a resistive load (your tungsten wire stop/turn signal lamps) to get to the ground potential of the steel body (and there by back to the battery). If it can not find a good electrical circuit to the front or rear of the car, it finds what ever it can. I assume the former owner some how attached your radio to the brake light circuit for power so when you turn on your signals it goes to ground by way of the radio.