The 22-243, in any of its varients, is a great cartridge and a blast to shoot. Unfortunately, you will never be able to realize its potential in a 22", 12 twist barrel.
A case that large really benefits from heavy bullets like the 75gr A-Max or 80gr VLD/HPBT bullets. Light bullets in large cases burn lots of powder for small returns in velocity. You may not even see that much velocity increase with light bullets in that short of a barrel.
I shoot a 22-243AI with a 28", 8 twist barrel. I get about 3,670 ft/sec with the 75gr A-Max and the accuracy is amazing. I have made many kills on groundhogs out to 900yds. This setup shoots extremely flat, resists wind well, destroys varmints and has low recoil. You could not get this type of performance from the light bullets at any attainable velocity in the 22-243 Mid.
If I were you I'd save the rechambering money and put it towards a fast twist .224 barrel. You'll be glad that you did.