Harry with Bella's advantages and obstacles would have taken over the world in the first few years.

You mean weeks, right?

I think Harry would probably have gotten himself killed the minute he wandered near Volterra and was too obtrusively dangerous.

He wouldn't have gone there until prepared to deal with them. The very first thing Harry would have done (or possibly second after being turned) would have been experimenting with Alice's precognition. It's at least three orders of magnitudes more powerful than all other abilities combined, can be experimented on safely and instantly and allows you to do most other experiments safely and instantly as well. Given a fully cooperative Alice and his creativity the only credible opponent at that point is the author (there are no wolves around at that point, the Volturi not aware of Half Vampires [right? I have no knowledge of canon], and the "free will" limitation doesn't make sense and either can be eliminated, mitigated, or has implications that completely change the picture, like that they are in a simulation that might be hackable, or characters in a story).