You need to extend the visibility of the last frame with your video editing software for the video file. Some video editors contain video filters that extend the visibility of single frames. Or they have the feature of including still images into a video and you can configure how long you want to show this still image. Not endless, but up to some minutes.
If you don't have such filters, you can append duplicates of the last frame for some minutes with your video editor, so the video is a few minutes longer and in the last minutes it displays the duplicated frame.
There is another workaround. Export your last frame to an image file. Add this image to OBS with an image source. Make the image source exactly the same size as the video source. Put the image source directly behind the video source. This way, the image is obscured by the video, as long as the video is running. If the video stops, it vanishes from display and the image behind it becomes visible. Since it is a copy of the last frame, it looks like as if the video stays on the last frame.