^Congrats! That was a very fast finish. I'm sure its faster than any UN diplo finish I've ever done at any difficulty level. So, you don't suck at diplo anymore. Quite the opposite! 

Emporer 1360AD win.
What would you do?
Here's what I did:
Hope you enjoyed the story.
Sun Tzu Wu
Yes, I enjoyed it! Very resourceful to get up to 4 LB's for defense so quickly. If I was Louis, I'd never attack 4LB's in a city with that stack, I'd always either bring along seige or HA's with the withdrawal promotions.
But if I were Louis and went after a capitol that had 1 archer, and then 3 turns later by the time I walk through the territory it had 4 LB's (organically grown, rather than re-stationed from nearby), I would shout some very foul language about how the AI cheats.
Hope your game continues to proceed well!
another option might have been your own cats and beat down his stack.
Did you think about making horses to flank his cats and take them out before they could kill your defense?
I have no idea if those ideas would have worked better or not, but it's what I thought of as alternatives.