The wiring really gets tricky with a 3 wire controller. I'm not even sure it can be done without at least adding some diodes to the wiring. The relay might help but I am not sure how that would factor into the system.
The limitation is that you want the brake light filament to also serve as the turn signal filament. Remember that the other filament has to serve as the taillight. You want the brake lights to come on whenever you press the brake whether or not you are using the turn signals. But you want the turn signal to interrupt the power to one of the brake lights when you use that direction's turn signal while braking.
Your turn signal switch will only send power to either the right or left turn signal when you move the switch to signal right or left. So you cannot wire the switch upstream or downstream of the brake light switch because it will stop the power flow to the brake light filament. If you wire it in parallel it will not flash the turn signal when you are using the brakes.
Your situation is even more complicated because you do not have a separate wire for the front and rear turn signals.
This is the reason they created 7 wire turn signal controllers. Those have an internal set of contacts that coordinate the turn signal/brake light function. I strongly suggest you save a lot of time and frustration and get one. It will be the best $30 you spend. Remember that the flasher has to match the voltage of the vehicle. Because you are using an 1157 bulb I assume you are converting to 12 volts.
7 Wire Turn Signal Switch