Most of your issue seems to be the MTB grips forcing the controls in further. What I'd suggest to complete the roadification process is to remove the grips.

Next is to find the best feeling spot to put the road drops. Mount then in a little from the ends and then go riding. Play with how far in or out you want them. You're just running bare bars at this point, no grips so some gloves may help with adding friction and padding to your hands. Work the ends until you're really happy with the feel of their position. At that point the bars from outer edge to outer edge should be somewhere around 40 to 45 cm across. Use a hacksaw to cut off the bits of flat bar sticking out.

You may also want to play with turning the bars upside down and run with them "dropped". Or as mentioned you can often find really light flat bars for cheap since they are generally out of style.

Finally once you're happy with the geometry of the setup put your brake levers as far in or out as you want to where you can get at them and they don't get in the way. When happy with all the control arrangements wrap the bars from the brake lever bases out and around the clamp on ends and then around the drops to the very end. Using bar tape removes any limitations on where you mount all the stuff since you just wrap around it however you want.

For the final touch if you're not riding on any dirt paths that may have some loose spots you can go even more narrow in the tire department. Panaracer Pasela 26 x 1.25 road tires roll like greased lightning and if you get the TG versions for a little extra money they are highly flat resistant as well.

A setup like it is about as fast as you'll go on a mountain bike.

Flat bars used shouldn't cost more than $20. Heck, if you were in Canada I'd send you one I've got for the cost of shipping. Bar "cork" tape is around another $10. If you opted for the Pasela or other really narrow road slick you're looking at around another $50 for the pair if you shop frugally. Finally something that may work better than narrower tires for building your road speed would be to dump the suspension forks in favour of some rigid forks. Look at both MTB and 29'er rigid forks with disc brake mounts. Either will work in your case since you're not worried about the rim size thanks to using discs. The rigid fork will bob less so more of your effort gets to the tire contact patch to go faster.

Of course all these mods is really limiting the ability to ride offroad on loose surfaces. It'll be fine for hard packed walking trails but as soon as you're faced with some rocky trail climbs and loose mud or sand you'll be back to wishing for knobbies and more upright bars. But if it's all road for you at this point I can see you getting this set up with a flat bar, bar tape and rigid forks for less than your $100 budget. Then later narrower tires for the final cherry on the sundae. But the real meat and potatoes here is the bar setup and going over to rigid forks. Oh, the rigid forks will be a little lighter than the suspension forks. And while I don't recomend doing it right away in case you change your mind about how you want to ride if you find you really love the road and don't do trails then you can sell the suspension fork to get most of your $100 back.