At our church, we occasionally display a live video on our screen. For example, if there is a puppet show, it makes the show more visible to the congregation. This is a secondary use of our video feed, however. Our main purpose is to live stream the services. So we use OBS Studio to manage the live video and audio. With OpenLP running, we have text overlays, image overlays, full-screen images and video, all managed by OpenLP.
In OBS, you can send any video feed, or the stream itself, to a monitor/projector and we send it to the same screen that OpenLP is using. To see the live fee, we use the "Show Desktop" button on the OpenLP live panel and the stream appears instantly. If we want to blank that or show an OpenLP item, we use the other buttons.
Our setup: a modest Windows PC with an SSD, three old iPhones running droidCam, a feed from our mixer to the PC, OBS Studio and the droidCam plugin for OBS. We manage everything with a somewhat complex AutoHotKey script that lets our operator switch scenes (with or without OpenLP overlays) and advance the OpenLP slides/verses using just the keypad on the PC. Cost: license for droidCam, hard drive & memory upgrade on the PC and cables. And a lot of time to get it all working. With AutoHotKey, we also mute/unmute inputs to our mixer.
Someone will have to be comfortable with OBS to get it working for you, but all the functions are explained well. It's not difficult to use. OBS, OpenLP and our mixer all have APIs that AutoHotKey can use to interact.