Help in converting Trek Domane 2.0 road bike to be more comfortable and commute-friendly
Regardless of the bar type you wind up with, the main way to raise the bars that much without swapping forks is via a steerer extender, which isn't safe to use with a carbon-steerered fork such as this, especially under a larger rider.

If you want the ability to go very tall, that means a steel steerer. There are some carbon forks with them that would wouldn't be a horrible mismatch with the bike that exist for this kind of purpose. Origin8 makes one. 

Opinions are a little mixed about the viability of getting the kind of height you're talking about out of a steerer extender on an al steerer. Usually manufacturers prescribe a spacer limit on such forks of no more than around 60mm and sometimes less, and I personally have never seen any reason why it makes sense to cheat that number using a steerer extender, as the resultant problems with excess leverage on the steerer will be the same. Again as a taller rider, I don't think this is a reasonable approach.

There is not a way to do any of this without re-cabling the bike unless the current cables and housings happen to all be way too long. Also, few if any of the bar types that would make sense for your situation use drop-bar type controls like you have, which is another major expense of switching. And, you're right to look in the direction of other bar types, because it's unlikely you're going to be able to make much use of the drops no matter what, at which point drop bars make little sense.

Unless you need to be on a road bike, I would look at getting something like an Elektra Townie, or a recumbent if you could find one where the seat ergonomics work.