A Little Change to Jax

I believe Jax is a good example of a simple kit, but it doesn't feel satisfying to play. I think this is caused by an bland passive and a bland ultimate. I would like to see some small things changed while keeping him simple and relatively the same. Disclaimer with these changes scalings and base damages and base stats will need to be looked at. Also minor lore changes may be needed but nothing huge. And I believe Riot will change Conqueror so that it doesn't work nearly the same next season which contributes to the ultimate change.

Changes his passive and make it his current ultimate passive. This would give Jax clear opportunity to trade early and for opponents to have an idea when he will probably engage.

Change is ultimate. Named something like "Now This is a Real Weapon" animation would change is lamp post into a real weapon and the ability would be something along the lines of 50% of his damage dealt over the next 5 seconds is true damage and basic attacking adds 1 second per basic attack to the duration capping at 10 total seconds.

My goal from those changes were to give home clear trading opportunities in lane and so he isn't so weak until level 6. Also to give him a ultimate that feels more rewarding. Also to make him less of a stat check champion. And lastly to force him to build more defensively when he gets to the later stages of the game.

What do you guys think? I would like to know if you guys could think this would work and maybe make him more unique and rewarding to play. Please let me know your thoughts. And be constructive if you don't like it tell me what about it you don't like.