If you look through the "What I would do differently thread", there are lots of posts regarding wishing they had more height.
I personally have (I have built 3 houses) always gone with the extra height in the basement. Keep in mind, a 8' wall pour, after adding in the concrete floor is only ~ 7'8".
My current house is 9' pour (~ 8'8"), and where my HVAC Duct Line & I-beam are that I also have a 'lower' riser, it definitely feels a bit tight - I don't think it would have been passable at 8' poured wall.
One option I have seen some people do, is having only a section of the basement 'dug deeper'... although, many of the ones I have seen like this, have been instead of a riser, but conceptually, you could do it for the whole room. I am a bit surprised at the quoted cost, that seems very high to me... I live in greater Cincinnati, and housing costs I thought were similar to St. Louis, and when I we had ours house built in 2012, our cost for a 2,850 square foot foundation was only about $3,500 (They said almost 8,000, but you get 50%+ off all options , so, closer to $3,500 in real cost).
I put in a Mini-Split to my theater room vs worrying about HVAC...although, still have to have an air exchange plan.
One thing you might look at, is what the I-beam & support poll locations are. In my theater area, they used a lighter gauge I-beam, so, they then needed a support poll. I have another span in my basement that is great and they used a thicker gauge I-beam to avoid yet another support pole... in retrospect, I wish I had looked at that/thought through it more in advance.
They also put my Electric Panel in my theater room, which compromises sound proofing some... to do it again, I would have had them put it somewhere else.