I have reviewed your requirement, Need the following things are mandatory in web site
1) SSL installed in Server
2) Server response time 90/sec.
Then we can use the WEB RTC protocol as you want, because every video will required to this protocol.
as you mentioned, you will not use any 3rd party, so i have to build application only for you. I hope you understand my point, may be it will take more time as i assume, but quality will be 100% accurate as you want.
I have 9 years of experience in PHP, Codeigniter, Yii and MySql
"Please ping me on chat and mention you comfortable time to discuss the task list.
I will give you 6 month free support after finish the application."