I wouldn't with drop bars and sti levers, that will cost big!
I myself would go with a lower stem then add bar ends. Bar ends put your hands in a postions similar to those of the hoods on a drop bar. With the lower stem, you won't be losing too much in aerodynamics vs drop bars. Not to mention most drop bar riding is done on the hoods. Not to mention you can slap on some cheap aerobars if you think you're losing too much of the aero position. I've known some riders with aerobars on MTBs. I wouldn't myself but it's got to help one way or another.
Go with the slick narrow tires, you'll be fine.
I've got my MTB set with lower bars than you. I tested myself on a flat pavement ride and averaged 18 mph for 20 miles with KNOBBIES. The bike you have would be just fine if you get into form and add the few advantages mentioned. Plus you have a rigid fork which is even better than shocks as far as efficiency riding on pavement.
First I'd lower the bar profile like so!
Notice the bar ends, they are excellent for climbing as well.
