When you say it dosnt start. Does it not crank? Just click? We need more info. What i would do is the normal procedure for not starting. Ride it and get it to the point that it dosnt run. Pull the plug and check for spark. Do a compression test. And try starting fluid to see if it pops off on that. Simple fuel spark and air test. 300ex's were known for boiling fuel in there metal gas tanks(if this is an older 300ex) didnt specify a year. Could be that simple. Welcome to the forums!
again, have you had this battery load tested or NOT??
as a GOOD battery should NOT go dead after running a while?
it should be able to handle a discharge for a pretty long time as this atv doesn't come OEM with a ton of electronic's on it to drain it that fast
if a NEW battery has bad cells, it will discharge FAST, the amount of juice sucked down to start it the first time, might be all it has
so, just cause its NEW< doesn;t mean its GOOD battery!
you need to rule it out 100% before moving forward here!
Check your battery. This is exactly what happened to my 300. Got a new battery due old would not stay charged. Same problem, Changed solinoid,starter, battery cables, problem continued. Figured had a short to frame somewhere in harness but finally did a load check on new battery, bingo...got me a better new battery. Problem solved.