I think I understand what you're trying to do, and the answer is no.
If you change the record settings to Lock to App, SR does keep recording when the app is minimized but Windows no longer updates the content so you're left with a static image. That's true even if the audio keeps playing, unfortunately.
Assuming there's no way to simply download your online training video, you'll either need to record it when you're not using the computer for other things (like when you're asleep) or else use a second monitor and set SR to record on the one you aren't using.