HiNice job bro'. I'm not hijacking this thread, just posting a solution for a similar problem I had.
I had a similar problem. The blower worked at all speeds but the A/C compressor started to come on/off (light blinked) at 4th speed. Then started doing that at 3rd speed. When it started doing that at 2nd speed, I checked the wiring diagram and noticed that with the A/C button on and the blower at any speed, a ground signal is fed to the control unit (behind right button). What I did was to ran a supplemental ground to the signal fed to the control unit. Everything went back to normal, cold air at any blower speed. 6 months later and still working.
I am helping my son with his Tiburon. He has AC only on fan speed 2. On setting 3, the AC compressor cuts in and out while the AC LED flickers . On setting 4, the AC light goes out and he just gets humid air blowing in.
I am trying to understand your fix. If I am reading correctly, you are saying that you added (soldered?) a wire from terminal 87 (pin #1?) in the temperature dial connector (for selecting cold vs. hot) to the wire leading to terminal 86? Is that correct? Also, did that prevent the car from blowing heated air when needed? Thanks for additional explanation and help!