Have you looked at what your script sends in a traffic analyzer like Wireshark LWP::Debug (as you're using HTTPS, it's likely easier to dump things at the source side)?

The error message is highly unclear about whether the Sharepoint server received something with Content-Type: application/json, and could not handle it, or whether it wants ti send something with Content-Type: application/json and your client doesn't offer that in the appropriate way.

I see that you're setting both, Accept and Content-Type headers, so the next thing would be to make really sure that they actually get sent over the wire.

Maybe you can find out if the Sharepoint logs point out a better error cause.