Possible to embed the jitsi-meet UI in a same-origin iframe to access participant video streams?

I am attempting to install my own Jitsi server for use by an educational website I maintain that wishes to have integrated videoconferencing. The Quick Install instructions appear to create a single server that includes the jitsi-meet and videobridge components, which can be embedded with the IFrame API:

                IFrame API
+-------------+ (JitsiMeetExternalAPI) +--------------+
| example.edu +----------+------------>+ jitsi-meet   |
+-------------+          |         443 | prosody/xmpp |
  example.edu            |             | jicofo       |
                         +------------>+ videobridge  |
                                  4443 +--------------+
                                 10000   jitsi.example.edu

However I wish to have my own fully custom UI that is integrated into my educational site. The IFrame API provides nearly everything I require to control a meeting, but does not provide a way to access the video streams. I want to be able to create a <video> element on my own site that can display the video and shared screen streams of participants in the meeting.

JitsiMeetExternalAPI does appear to provide an undocumented API _getParticipantVideo(participantId) which returns the HTMLVideoElement inside the iframe for the participant video stream. From that video element I could steal its srcObject MediaStream and copy it to my own video element outside the iframe. However the _getParticipantVideo() function only works if the iframe is same-origin.

So I think I want to alter my environment such that the jitsi-meet web UI is moved into the same domain as the site trying to embed it:

+-------------+ ???                    +--------------+
| example.edu +----------+------------>+              |
| jitsi-meet  |          |         ??? | prosody/xmpp |
+-------------+          |             | jicofo       |
  example.edu            +------------>+ videobridge  |
                                  4443 +--------------+
                                 10000   jitsi.example.edu

Is it reasonably easy to separate out the jitsi-meet component from the other components on the Jitsi server in this way? Is there documentation to do this?

you could disable same-origin header on nginx side