How about this as a solution:
When Katarina activates her W, her skills gain a special bonus.
Her Q now deals max damage for each bounce and applies healing reduction at the cost of no longer applying marks. 1/3 the damage the marks would have ealt if exploded are instead dealt up front.
Her E now grants 30% damage reduction for 3 seconds (also in this version, her normal shunpo gives no damage reduction normally, just like before the rework).
If you activate W again, you get the current W with maybe some reduced scaling or base damage.
When she uses W and then either Q,W, or E, W goes on a cooldown dependent on the imbued skill. So WQ would leave Q and W on a similar cooldown, while WW would leave W on a much shorter cooldown.
What this does:
It allows katarina to actually have good ranged harass in lane like she used to. And to get that good harass, it means she must give up her W. When she gets the full damage on her Q, it means she imbued it with her W and cannot follow up with sinister steel because it is now on cooldown.
If you want to do the current Q then jump in and sinister steel, you can make that choice but you wouldn't get the 30% damage reduction as you used W for sinister steel instead of on your shunpo.
I'd like to stress against that Q would only mark if it was not imbued, meaning it still falls off in damage for each target hit and would not apply grievous wounds. If you follow this path, it means sinister steel is still open and you would benefit from going in and popping those marks. This preserves her ability to go all in someone, while at the same time allowing her to have ranged harass if she chooses. And the way it's designed to make W go on cooldown when you use it with Q means that you can't have both ranged and close-up superiority at the same time, you have to choose one depending on the situation. You also would have to choose the damage reduction for shunpo in situations that required it. Because it's not on every use and requires decision making as well as the sacrifice of other useful ability combinations, the % of damage reduction would be increased to 30% (still less than what it was pre rework).
Considering sinister steel has no cast time anyway, mashing WW when you want it would have a slight if not no delay in comparison to the old version. Making you press W twice to get sinister steel would barely affect the spell itself, but it would allow the other spells to be used like old killer instincts.
I think her ult might need to be buffed a little more considering sinister steel would be slightly nerfed in damage and not be available nearly as much as you'd have to pick it over the other 2 spell bonuses, making katarina slightly more like her reverted self but still with decent farming capability.