JSONP can only use GET since it uses the script tags instead of XHR. The JS SDK sets the method in the _method query param. If it’s not working, it’s a problem with the API not accepting it.
i.e. the actual request is something along the lines of what’s below, which works for me.
GET /kraken/users/:user/follows/channels/:channel?callback=something&_method=PUT&oauth_token=some_valid_token