Raising the Boom
Another possible solution might be to raise the boom 6" to 12" to make more room for the bimini. The loss of sail area would probably not be very noticable. It could be done with a very short reef. That way it could be taken out fairly easially when the bimimi is not wanted, like for racing. Raising the boom 6" would make enough room to sit on top of the coaming under the bimini. Standing headroom would probably require raising the boom 18" or so. 6 to 12 seems like a good compromise. The picture below is of a nicely done bimini installation with the boom raised and the OEM mainsheet location.
Moving the Traveler Forward
Moving the traveler to the front of the cockpit would clear a lot of room for a bimini. The way I have drawn it there is about 45" from the bimini to the seats and 60" to the cockpit floor. I have also left room at the front to allow easier movement from the cockpit to the side decks. The bimini could extend further aft with a slot for the backstay to go through. I would want to be able to stand up at the very back of the cockpit behind the bimini. Maybe moving the traveler forward and raising the boom 6" would be the best set up. There are pros and cons to having the traveler at the front of the cockpit. I have made a list on my traveler page.
I don't know if I'll ever add a bimini. When you sit on the boat and think about how the bimini would fit it seems pretty cramped. A better solution for us might be a big umbrella. But if we were in Texas I might think differently. Having shade while sailing can be a big plus.