What would you do?

Try some EQ

I want to increase soundstage and curb a slight tendency for high end harshness.

Acoustic treatment of the room would specifically address both of those desiderata.

And I know that the room plays a big role but I can't make changes there.

What does this mean anyway?
I can't move the system into another room.
I can't knock down a wall.
I'm not permitted to pull the speakers 3' out into the room.
I'm not permitted to move the furniture around.
I'm not permitted to hang 4" deep panels in certain places on the walls.

I agree w/Jones. Get a Schitt Loki, a 4 band tone control which will help you tailor your sound issues. It has bands at 400 and 2000hz which might easily help correct your problem. You put it between your source and your integrated amp. It is relatively benign and has a pass-by switch. It's cheap (@$150) and easily obtainable. I use one in a dedicated headphone system and really appreciate it on a lot of my CD's. (I don't need it in my main system  but I tried it there an it works well). FWIW, if your integrated has a tape loop that is a perfect place for this unit. 

BTW, I assume you have tried already but if not - adjust your speaker's  toe in until the axis crosses well in front  of your listening chair. That works for many set ups. 

Let us know how how the magic works out.

Thanks for all your suggestions. Newbee, I cannot, in fact,  re-arrange anything except I have always had  the speakers out from the wall.  The equalizer sound interesting. While it might take any harshness away, I guess (?) it would not do anything to the soundstage.  While I have tried angling the speakers toward the listening position, I find the soundstage better when there is no angle at all (apparently Jeff Joseph suggested this). But I can try again. Thanks again.