After talking with you about AP bruisers, I've started playing a lot of Shyvana top. From what i hear, I'd say these would be a decent start at least.:
New Passive: Burnout: Whenever you use an ability, Shyvana surrounds herself in flames for 3 seconds and deals 10-70(+20% Bonus AD)(based on level) magic damage to surrounding enemies each second. Basic attacking an enemy while Burnout is active deals 25% of Burnout's damage to all enemies surrounding Shyvana. Additionally, Shyvana deals 10% bonus magic damage to dragons and to Vilemaw.
New W: Harden Scales: Shyvana become invulnerable and slowed by 40% for 0.75 seconds. Afterwards, she gains 15/20/25/30/35% bonus movement speed for 7 seconds. If she was hit by an attack or ability while invulnerable, she gain 5/10/15/20/25 bonus armor and magic resistance for 15 seconds. If Shyvana was hit by an immobilizing effect while invulnerable, she gain double the movement speed instead.
R: is no longer unstoppable, but can still activate W during the dive.
I think lore wise the best change would to be a simpler change if she can transform when she gets angry and gives into her draconic side. That would be a perfect match for gameplay and also a bit more interesting as she has to learn to practive temperance in a country where most hate her. Could be kinda fun to explore since her only friend there (and the world for that matter) is Jarvan and probably the only one she can express these frustrations with. Plus i ship those two so there is that hahahahaha.
As for her current state, This is the build i've been using:

Gives her a lot of utility in a team fight and i was very surprised at how bulky i could get with Grasp of the Undying and Overgrowth added on top. I'll also add that after i get 1 or 2 kills, its amazing the number of people who suicide into me. I have a hard time imagining they'd play that way into a 2/0 Illaoi, but they do anyways. Really, they need to practice a lot more careful trades, since i get a TON of free damage if they do it wrong. Like i played a xin Zhao that always traded with his nock up down. He should have been stacking it, dashing onto me, nock up, combo, retreat behind minions. Instead i beat him down for playing sloppy again and again. I'll say i dislike how easy that is for me, but thats why i created the changes i mentioned above, at least make a sigum of skill required.