They said that when you get sick in the first 6 weeks, the insurance company pays. They also said it was my responsibility to fight with the insurance company to get that money.

That's actually true.

I contacted HR and told them that they need to submit this documentation to AOK so that I can get paid. Now they are stonewalling me, saying only "We never got a request for any documents from AOK." I reply, "Please CALL AOK and ASK THEM how to solve this problem." They then reply "We never got any request for documents from AOK".

That's not how that works. You get the document from your employer and send it to the AOK.

(they agreed to take the Kaution spread out over 3 monthly installments)

As is the law.

TLDR: Shady ass company refuses to submit a document to AOK which would permit me to get paid for sick time. Which government agency can I turn to for help?

The local court who will allow you to sue the company so that the company must give the document to you.