Overall in my opinion his lategame should be buffed but thats just me.

Anyways I am having a real difficulty as lategame lee, I build him correctly and stuff its just how I play. I have trouble actually doing my role and killing the enemy carries most of the time.

How do you make him theoretically strong lategame what do you do to counter his fall off?

He is really strong early/mid game. His balance is a weaker late game.

Take advantage of your strength and snowball yourself.

He is either a straight glass cannon or a support/tank late game. If you're a mid Lee, then glass cannon. If you're a jungler, then you're probably going to build Runic Bulwark at some point. If you're top transition depending on how well you snowballed or depending on your team's need.

His damage is fine with Black Cleaver+Bloodthirster+Last Whisper, but he will go squish pretty fast.

Imo Lee only needs 2 items to do great damge early-mid game. Any combo of Brut>BC and Wriggles Mallet Hexdrinker or BT. I would then seriously consider getting resistances. If you don't get fed Lee still makes for a great peeler/anti engage/chaser.