Hi,
if I understand what you’re trying to do, you actually can loopOut time by pre-composing, then enabling time remapping and setting two keyframes corresponding to the start and ending points of your animation.
Once you have those you can use an expression to loop the whole thing.
You might need to set the second keyframe one frame before the actual end of the comp to ensure it loops out nice and even, ’cause sometimes it flutters on the final frame if you don’t.
Hope this helps!
EDIT:
for the last part, I mean set the end keyframe, then copy it and past it one frame before. Then delete the other.