What you are looking for is a mixer, like the Black Magic ATEM. This type of gear is expensive though and usually only used by TV-studios and professional broadcasting services, so I would think twice before buying one.
An alternative is hard to come by, since you need HMDI or (preferably) SDI inputs that get mixed, which you cannot simply do on a PC or Mac.
A quick google search gave me dozens of options. You can use yellowduck for example. Here's a tutorial on how to use it. An alternative is promovgram. There really are a lot of options, so I would recommend just googling yourself.
First: 4k for livestreaming is pretty overkill. Please remember that your streaming device (your computer) and internet upload speed need handle this... AND to top things of not all of your viewers will be able to watch the stream in 4k because of their internet and missing out 4k devices.
Alpha Gaming (Harris Heller) is covering everything you want to know ...
At best, you can use the software Handbrake
It greatly reduces the video volume and has little effect on the quality. In this software, make changes to the resolution section and other items. I'm sure if you test a few times you will notice its very good performance and this software is free.
I see 2 things you are trying to achieve here:
video is private, but becomes public. at api.video, we have the concept of private videos - where users must have a one time token to watch. Anyone else using that token will be unable to watch the video. Once you reach your view threshold, you can make the video public - removing this token behavior.