Lag made it hard to punish, but you used smashes WAY too much. Notice how when you went for the kill, it was easy to predict.
Smashes are kill moves, not damage-rackers. That's why it took awhile until you could actually kill me. When you use a move in Brawl in succession, it decays, meaning it loses its killing potency.
Also, I shouldn't be able to outcamp you as your own character. When you're using TL, you should always try and keep your opponent away with projectiles.
With Lucas, abusing smashes is never a good idea. Lucas' smashes are vital. His early KO potential is what keeps him from being such a bad character, and if you let your smashes decay, then you've taken out one of his greatest assets. Also, grounded PKF has ending lag that can be punished, that's why you SH it. I couldn't really show you how to properly keep your opponent away with PKF since boomerang practically destroys my deffensive game.
Other than that, you're doing okay. Just work on what I've told you.