I'm thinking the same thing 2 door is... A bad magneto coil will fial intermittantly as well as a bad CDI unit. They'll run fine until the warm up and then start to cut out... usually on one second then off, let the engine rest for about 5 minutes and it fires right back up.
It's also very possible you got a bad CDI or mag coil in your new engine kit as I've had this problem before with a new Dax engine. I just got it built and installed, went to start it and it outright refused... checked for fuel and there was fuel in my carb so I pulled the plug, grounded it and pulled my start cord a few times... nothing. I was suspecting my new CDI so I put my old one back on and again, Nothing.
Next I did an ohm check on the blue and black wires where it measured dead open. Took the mag coil out and put in a known good one, connected it to the old CDI and it fired right up.
Check your mag coil by measuring the resistance between the black and the blue wire, it should be around 560 ohms, then black to ground (any bare metal on the engine) should read 0 ohms. If your blue wire is open or the resistance is really low, or zero resistance then the mag coil is bad. You may need to carry the meter with you and measure it right after it dies to catch it.
If the mag coil checks good, then check your CDI unit. I forgot what it should read, but hopefully someone else in here can help you with that part... unless you have another one that you know is good, then swap it out and see if the problem goes away. The mag coils don't like water or moisture, if water gets in there, you can pretty much kiss it goodbye. The CDI unit gets ruined by overtorquing the bolts holding it on, most the time one of the ears just breaks off, but it's potted in solid plastic so it can easily suffer internal damage. It also doesn't like vibration so solid mounting it to the frame is also a bad idea, a piece of rubber froman inner tube between the CDI unit and the frame will be enough to protect it as well as mounting it with a zip tie or 3 instead of the bracket.
One more trouble area in the ignition system is the stock spark plug boot, it can come loose or break internally and then the spark has to jump that gap plus the plug gap so it may or may not fire the plug it the stock boot went bad. Now to add more insult to injury here, if the boot is bad or broken and the spark has to jump this extra gap in there, it puts stress on the cdi and can cause the CDI to fail.
Also if you unscrew the boot off the wire and unscrew the wire off the CDI unit, cut off about 1/2" from each side and screw it all back together you'll get a better contact there.
As far as the engine being hot... Yes, these things run very hot... 280 to 320 degrees is about normal with 320 or so being right after a really hard run, with a new engine that's not fully broken in, it'll run hotter too, that's why it's not recommended to ride more than about 30 minutes then let it cool. Best way to measure the temperature is with an infrared "point and shoot" type thermometer or a cylinder head temp gauge with the probe on the spark plug base. Keep your operating temp well below 400 degrees because that's where the aluminum starts to lose it's strength... 320 to 360 is ok for short bursts, but above 360 should be avoided.
Hope this helps.