If looking for HTTP paths, when the API call to Brightcove is correct you won't see the rtmp:// urls.
Since you're getting the rtmp URLs, this verifies you're using an API token with URL access, which is good. A request like this should return the playlist and the http URLs (insert your token and playlist ID).
http://api.brightcove.com/services/library?command=find_playlist_by_id&token={yourToken}&playlist_id={yourPlaylist}&video_fields=FLVURL&media_delivery=http
This API test tool can help build the queries for you, and show the expected results: http://opensource.brightcove.com/tool/api-test-tool
I'm not seeing what would be wrong in your code, but in case you haven't tried this already, debugging in the browser can help you confirm the API results being returned, without having to access it via code. This help you root out any issues with the code you're using to access the values, vs problems with the values themselves. This is an overview on step-debugging in Chrome if you haven't used this before: https://developers.google.com/chrome-developer-tools/docs/scripts-breakpoints