I would check the basics first, like battery terminals tight.
Before I bought a stator I would check the charge rate at the battery with a DC volt meter. if that is not at 14.5v then I would check alternator output with an AC volt meter, rectifier/regulator with an ohmmeter, connectors, fuses, etc. instead of throwing $400 chunks at it.
If you were riding at low rpm it may not make enough to sustain battery charge, but over about 3k rpm should be making voltage.
I don't know how much ampherage those electric items draw. I think the klr has a pretty healthy output though and should handle it.
There are stator hopper-uppers out there such as Baja Designs of which I am a dealer.
'72 BMW R75/5 (Col. Klink),
'02 KTM 400 E/XC,
'05 ktm 125 sx (for sale)
"Hey, are you okay? Can you get up?
I thought you said you could ride"
All the best