Bummer...
It does sound like carb... a jet may be plugged, or the needle slide is sticking... She needs a cleaning...
The loud click,... is a decompression relay, (it makes the engine easier to turn over at start by lifting the exhaust valve for a fraction of a second)... That's a normal sound...
Now,.. why your battery is dead, is another can o' worms... but,... since you replaced it recently, let's just check that battery connections are good.... (a loose connection will affect charging and starting)( also, check the ground strap connection at the rear of the engine near the clutch actuator)...
... and lastly,... (new to bikes?),... wear jeans.. (they don't melt)...
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