OP, I will apologize to you for all these illiterate Freds. You said you didn't want clip on TT bars and then they all tell you to get clip on TT bars. And Spinacci. #foreheadsmack.
Yes the flat top aero road bar is an excellent idea for you. Something like a Specialized Aerofly would be good. They are meant to be bare on the tops, but in your case you could consider taping the tops and putting an extra layer of padding where your forearms rest. I would think that would be vastly more comfortable than a round bar under your arms for miles and miles. I have used this handlebar extensively and it accommodates your position well.
The only downside to these flat top aero road bars as that you want to keep the tops horizontal. That may not suit your preferred bar angle at the hoods. I could never quite get Dura Ace hoods to be in the exact right spot for my hands/wrists on Aeroflys.
In defense of us Freds, the OP seemed to be resistant to the idea of a clip-on due to the looks and because he rides big hills (so presumably he doesn't want excess weight). Several of us suggested mini
clip-ons which would provide an aero hand hold but with less intrusive looks and minimal weight gain when compared to a full on Ironman distance tri-setup. Without UCI mass start regs to worry about (the OP doesn't mention racing) it is well worth considering clip ons as you only see pros using the Invisible Aero Bar position when they want to get aero but have no other option due to being in a mass start race. As soon as they get into a TT, out come the clip-ons.
And I will reinforce that even if going for a flat top bar, you need tape or some other sort of padding on the tops when riding in the IAB position as sweaty forearms and gloss finish carbon are a flat out dangerous combo, so there will need to be some sort of compromise made in aesthetics anyway in order to add grip.