Hi RodB,
Welcome to the forum, I'm from South Africa.
You can carbonate in the mini-kegs the same as you do in bottles, just needing less sugar, You just need to calculate the right amount of sugar or carbonation drops to add into the mini-keg. The gray plug with the hard plastic insert is able to withstand normal carbonation pressures without a problem.
Beer Priming Sugar Calculator | Brewer's Friend is a good calculator to use. Here is a thread on natural carbonation in kegs in general with some info on how much sugar to use.
https://www.homebrewtalk.com/threads/naturally-carbonating-in-keg.277566/
Couple of threads on the mini kegs.
https://www.homebrewtalk.com/threads/easy-kegs-and-mini-kegs.678913/https://www.homebrewtalk.com/threads/5l-mini-keg-with-tap-how-long-does-it-last.559463/
As you found out with the first attempt, putting already carbonated beer into a mini-keg is difficult. It will degas until an equilibrium pressure is reached, so either you have to over-carbonate before transferring into the mini-keg or try and pressurize the keg so the beer doesn't de-gas. The little CO2 cartridges aren't good enough for that, only for serving already carbonated beer in my opinion.
Force-carbing a mini-keg with a wand might work out, you just have to be really quick in pulling the wand and inserting the tap. If the beer is cold, it won't fizz out immediately, so you'll only lose the pressure in the headspace if you're quick enough and that you can replace from the little CO2 cartridge.
I'd try natural carbonation in the mini-keg first before trying to fiddle a wand.