Won’t start

The display will dim or blink when you hit the start button if the battery is weak.
You probaly just need a new battery. The bike starts off the battery and may have enough amps to start by pull start but using the starter may draw the amperage too much to start.
When you say "catches" that throws us off to starter gears. Do you mean turn over?
Got a video of it?

Not saying it isn't possible but I haven't heard of a PGM sometimes working and sometimes not on these newer bikes unless the plugs to it aren't in good but I've heard of a solenoid doing that.
I kinda want to commit to saying that the PGM is ok because it starts with a pull rope. I would start with a new battery or you'd need to at least test it under load which any parts house would do for ya.
Or jump it from your car directly to the atv battery touching the terminal connections and if it starts right then it's your battery. If it doesn't then jump it the opposite pole on the solenoid and see if it starts right if it does then it's your solenoid.
.... and no ... your bike won't start because the charger is hooked up unless it has It an engine start feature otherwise it only outputs usually 2, 10 or 15 amps depending on the type of battery.... in your case should only be 2.
And by the way, welcome to the forum!