Welcome to the site !!
From what you described , it sounds like either a bad Stator or Flywheel magnet , or possibly a bad Ground or connection OR , a bulb output that is greater than the capacity of the Lighting Coil output in the Stator ......
You need to determine the brand of aftermarket Stator and light kit (hopefully they are both the same brand and are compatible with each other) , next you need to confirm ALL connections are good (not loose) and that all Grounds are secure and tight , as a bad ground will backfeed thru the system and cause all sorts of symptoms ..
As far as your "Checklist" ...... I would take some time and go over ALL bearings , which include the Steering Head bearings , the Swingarm bearings , and Suspension Linkage and Wheel bearings .... This is a must do , and should be done EVERY season before riding .... Along with checking the valves (do that now) as well as inspect the clutch Plates for wear and the Basket for grooving .... , the 04 should have enough power to pull the Snow Bike kit in stock form .....
As far as the "chattering" goes , I would suspect it is either the Timing Chain and/or the Tensioner causing the noise you are hearing (there should not be any vibration along with it) , if there is a distinct vibration , I would refer back to the Flywheel as a source , possibly a Magnet coming loose throwing the flywheel out of balance ..... DO NOT attempt to remove the flywheel without the CORRECT flywheel puller made for your bike (even different CRF models use a different puller like the X and R) and make sure you use it correctly , if you crush the brass flare that is on the end of the crankshaft , you will ruin the crank and need to replace it , as you will lose oiling to the engine if it gets damaged , the little cap that goes over the brass end at the end of the threads that come with the Flywheel Puller needs to be put on lightly and not bottomed out , screw it on all the way , but then back it off slightly so not pressure is put on the brass flare , leave a slight gap between it and the brass piece so it is not crushed when tightening down the puller
As far as Battery and E-Start , the E-Start is not going to happen and X cases will not fit in your frame unless you have the frame cradle modified , and even then , it would be cheaper and better to just get a 450X , but you can use (if the kit is equipped for it) a battery to help power the lights , but the kit must be made for one , don't just install one and try to hook it up to your electrical or you will fry your electrical system ..... the battery it will most likely use is similar to a RC Car battery (6 C-cell Ni-Cad batteries)....but check the schematic for the aftermarket Stator and Light system to make sure it will accept one ....
Also , you do not really need a lighting kit , so there is the option of just removing the lights or just that kit if you can't get it working right , and just get a new one from Trail Tech or similar place that has one made specifically for your bike and needs (output amps , etc.) if you do really want/need the lights to work for evening trail riding and or snow riding getting back late to camp ..
Hope that helped some ....