If you're using OBS (which is what I use), set LiveSplits for Window Capture and it'll appear on the display. You can crop it if necessary by selecting the capture on OBS and adding the filter Crop/Pad.
I'm not well versed with any other streaming programs, but generally a display/window capture option should allow LiveSplits to be visible when you record or stream.
If you want LiveSplit to show with transparent background on your stream, here's how to do it: --right-click on LiveSplit (under OBS Sources) and select Filters. --Click on the + symbol on the bottom left corner of the pop-up screen and choose Color Key. --Click on Select Color and Pick Screen Color. Now you can select one of the background parts of LiveSplit. --You'll need to add as many color keys as necessary to remove all the background colors (you should see them disappear on the OBS preview screen).
Before doing this, you can change the background colors in LiveSplit to a single color and OBS should take care of all the colors in 1 or 2 color keys.
