So long range machine gun Kog failed
And Riot has decided that artillery Kog is 'unhealthy' and 'painful to play against' so that's out, too.
Now, Kog has always had three identities all competing in his kit. Long range machine gun and artillery platform were two of them.
The third? Immobile tank-buster. A sort of anti-Juggernaut packed to the gills with high base damage and resistance shred. So since Riot clearly feels the other two identities are off the table, why not focus on Kog's identity as an annihilator of hard targets? The only competition Kog has on that front is Vayne, and with her massive AD steroid and mobility she's more assassin-y than strictly tank-buster-y.
I think keeping Kog's 'kite and use trinity procs' gameplay pattern would be a good thing to keep. It's engaging and fun (quite a bit more interactive than his 'press W to win' remake). Unfortunately, Kog's new Ult doesn't really support this pattern nearly as well as it once did. The higher mana cost, longer stack timer, and pitiful damage against healthy targets are all quite punishing. Honestly, given Riot's apparent direction on Kog his Ult really doesn't make a great deal of sense on him anymore.
At this point it would probably be a good idea to just axe Kog's Ult and rebuild it from the ground up (as a Kog main since before Season 1, I would be sad to see it go. But the thing is already dead. Time to put it out of its misery). Keeping the Trinity fodder aspect is a good idea, so it would need to be spammable. That's an engaging part of Kog's gameplay pattern. Tweaking the ability so that it's better suited for ruining the enemy front line is probably a good idea, too, since that's the only game-play hook Kog really has left. That could be done pretty easily with a %Missing HP component. We would want the ability to flow seamlessly into Kog's Trinity use if at all possible. Right now, Kog's Ult interrupts both Kog's basic attacks and his movement, which serves to throw off his cadence. If the new Ult were either an attack modifier or didn't stutter Kog's movement then that would help Koggy do his job.
Beyond those three points (spammy, %Missing HP aspect, doesn't interrupt Kog's movement) it could be most anything. Aside, of course, from anything super-long range. The simplest option would probably be an attack modifier that gets more frightening the more times Kog hits the same target with it. But regardless of what Kog's Ult gets turned into, I really don't feel the one he's got now fits him anymore.