You always seem to find interesting things to work with!
My Mk1 Mini and my wife's Triumph GT6 don't have hazard lights at all. Our Spitfire has a very different switch so I am only familiar with the Mini switch through this message board.
Give me a little time to dig up some images and wiring diagrams. Don't be upset if they are for MGs or Triumphs. The basic switch function is the same.
EDIT2: Corey, I thought the wiring would be the same for a '74 MGB and your Mini. I was wrong. I cannot paste an image from the wiring diagrams I have at work so I will have to post a description. Maybe I'll find a wiring diagram to post later.
To work like the factory hazard switch you will need a minimum of 5 terminals. Use your multimeter to find the following combination/groups of terminals. With the switch OFF you should find two terminals that are connected (continuity between two terminals). Write down which those are. Now turn the switch ON and repeat your measurements looking for three (3) terminals that are tied together. The "2" terminals may/should be marked 1 & 2. The "3" terminals should/may be marked 5, 6, & 3.
Your harness should have a pair of light-green/brown wires near the switch. Connect those to the 2 switch terminals that have continuity when the switch is OFF. You should have three remaining wires, light-green/pink, green/white, and green/red. Connect these to the 3 terminals that have continuity to each other when the switch is ON.
So what does this do? The light-green/brown wires are coming from the turn signal flasher going to the turn signal switch. They pass through the hazard switch on the terminals that may be marked 1 & 2. When you turn the hazards on contacts 1 & 2 open and disable the turn signals. The light-green/pink wire is coming from the hazard flasher unit. If the switch terminals are marked, it goes on terminal 5. When the hazard switch is on, terminals 3, 5, & 6 are joined. This connects the hazard flasher to both the green/white and green/red wires which go to the turn signal lights on the right and left side of the car.
EDIT3: See the PDF linked below for the Canadian Mini 1000
http://www.minimania...Fs/wiring10.pdf
Look for item 154 (hazard flasher located near the coil, top left of schematic). Follow the wire from the flasher's "L" terminal down to item 153 (hazard switch). In this diagram, the hazard switch contacts make/break the turn signal flasher's input power (green wire) instead of light-green/brown wire. Either will work. Use what is correct for the wires in your harness.
Edited by dklawson, 23 March 2016 - 03:19 PM.