I would suggest actually just trying to install a new wiring harness that has the right signals. Plenty of cars of that style didn't even have a factory wiring harness for a trailer. Whoever installed the trailer hitch would also install the wiring harness, tapping into existing taillight wiring.
You will need some sort of timer circuit to do this without getting access to the separate wires inside the car. Technically you could probably do this with relays and RC timers, but you are going to have a much better time doing it with digital logic.
I would guess what you need to do is that at the falling edge of the "brake" signal, you set a timer. If the brake goes high before the timer runs out, you instead light the turn signals. Thus, when you turn on the hazards, the first flash will be the brake lights but the second will be the turn signals. You will also want a timer where if the brake signal is on for more than X milliseconds, you switch back from hazards to brake lights. That will make sure that if you e.g., tap the break and then press and hold the break you won't be displaying hazard lights. An improvement would be to add a counter to the hazard detection: only after three timed flashes do you switch over to the hazard lights to reduce the possibility of false triggering.
You could do this with discrete logic but I would recommend a small microcontroller. It might seem silly to use a micro for such a simple purpose but it may well be cheaper than a pair of 555s, a handful of NAND/NOT/OR gates and a small counter. Of course the silicon cost is probably going to be negligible anyway for this application: your power switches, connectors, and weatherproof enclosure are going to be more expensive.
You can use logic level MOSFETs or DC capable SSRs to drive the outputs and opto-isolators on the inputs to protect the logic.