It's referred to as a "Borderless Window" instead of "Fullscreen". Basically it's the same as windowed mode but it is set to fill the screen and the titlebar/borders are disabled. This allows you to tab out. I am under the impression that Borderless Window is, like regular windowed mode (since it's essentially the same), slightly slower than fullscreen since it has to draw the desktop.
In SDL_CreateWindow, remove the SDL_WINDOW_FULLSCREEN* flag(s) and add the SDL_WINDOW_BORDERLESS flag. Then set the window size to be the resolution of the desktop, I think.
Hmm... With AppNap do you think it would be possible to gain back that speed? Worth a try, I'd say!