First, dirty gas from switching to reserve or running very low (install a filter and clean the carb a bit), fuel cap not venting (when it tries to die remove the cap and see what happens), carb vents plugged (look at the drain lines on it for blockage), kickstand switch malfunctioning (less likely but just spray some contact cleaner in it or disconnect it). The TW200 (at least the older ones) should be able to run without a battery at all, so dying while the engine was on isn't likely a battery issue. I sure hope you charged the battery before you installed it though.
Dying when you pull the clutch is usually a fuel issue, which would make sense given your description. I would pull the bowl off the carb (read the stickies on here) and give a light cleaning to it and the jets and see if anything improves. Also, with the bowl off, hold the float up, turn the fuel on momentarily, then drop the float and make sure a little gas starts pouring. I forget how hard it is to remove the bowl with the carb on the bike, but I do recall it being doable.
If you've never before, get yourself four replacement screws (preferable SS allen heads) for the bowl. Helps a lot if you're removing it a lot as the stock ones strip easy. You can learn the details of that on here.
Also, since you'll be doing carb work here and there (Or you pay somebody a stupid amount of money to do it. I don't judge.) Search for the JIS driver thing. It'll save you a couple more stripped screws.