Ok, looks like I have the same cooler.
I'm not sure what different setups you have tried, but here's my thoughts.
Obviously we are going to pull the existing drain plug out. I'm not exactly sure what size hole it is going to leave, but I'm guessing that it is going to be somewhere close to 3/4"-7/8". I would get a 3/4" threaded brass pipe nipple, that is at least 1" longer than the hole. That should leave you with approx 1/2"-5/8" of threads on either side. On the outside all you need is a 3/4" flat washer and then your ball valve. For the inside, it looks like the "well" that the drain sits in is not going to be wide/flat enough to allow a full sized 3/4" stainless flat washer, which could be the cause of your leaks. I would find a 1/16"-1/8" thick 3/4" neoprene rubber washer. You will probably have to stretch it over the end of the pipe nipple as the OD of the nipple is prolly close to 7/8", but the tight fit is good. Outside the rubber washer you will need a 3/4" stainless steel washer. You may need to dremel out the inside of the washer slightly to get it to fit over the nipple, and looking at the width of the "well" that the drain sits in, you might need to cut down the outside of the washer on the sides to get it to lay flat against the cooler. Once you have ensured that everything can lay flat against the inside of the cooler, remove the pipe nipple from the cooler.
I used a 3/4" to 1/2" female IPT reducer to secure the nipple on the inside and the ball valve on the outside. I assembled the components that go on the inside of the cooler first. Rubber washer -> stainless steel washer -> copper reducer, using Teflon tape when screwing the reducer onto the pipe nipple. DON'T completely tighten the reducer onto the nipple, just finger tight. Prior to inserting the pipe nipple through the drain hole, I put a 1/4" - 3/8" bead of clear silicon caulking where the neoprene washer joins the pipe nipple. I did this to help seal the cooler drain hole cause it was just a little bigger than the pipe nipple. Between the rubber washer and silicone, you should have (I do) a good, water tight seal. Now you can assemble the outside of the cooler. Place the 3/4" flat washer over the pipe nipple (may have to dremel it out a little and doesn't have to be stainless, zinc coated will work just fine), put some Teflon tape on the threads of the pipe nipple and screw on your ball valve. You can hand tighten the ball valve onto the pipe nipple and maybe go a 1/2 to 3/4 turn with a wrench. You should still be able to rotate the valve and place the valve handle where you want it. Hold the ball valve assembly with a crescent wrench and then use another wrench to tighten the copper reducer. This way your ball valve stays where you want it, but you pull everything together. Tighten until you start to see a small amount of flex in the walls of the cooler and you should be good.
Sorry if this seems like a novel.... Just trying to explain how I did mine and would do a similar one in your style cooler. Hope it helps.