the barrel has to be set-back for a .243 Win' to clean-up when re-chambering to 6mm Rem'. it's a great change for a bolt gun; it's not an easy job with a Handi'. the 6mm Rem' Improved (of one variant or another) should be doable, however. even so, the use of factory-loaded 6mm' Rem' would not be an option in this new chamber because there would be no 'crush' built into it like there is with the setback of a bolt action barrel.........
you cannot take standard 6mm Rem' dies and use them to necksize an Improved 6mm because the dies are too narrow at the shoulder,etc.etc...... your standard 6mm dies would have to be opened up with the same reamer that is used for the chamber. BUT, that gives you one heck of a set-up with a die that is cut by the same reamer used on the chamber. the tightness of the neck may not be changeable in the new chamber unless going with longer brass like the .30-06, but the headspace and the shoulder-setback built into the die would make for long-lived brass (if the brass is good to start with). it'll cost you to have both the chamber and the die done; but you may wind up with one of the most accurate rifles around.
if you achieve this re-chambering there is still potential for one more significant problem if you use .30-06 brass to achieve a custom fit of the case neck in the chamber........the dreaded "donut" will occur with this setup. ....and outside neck turning to achieve case neck thickness that is consistent will probably be necessary!
www.redding-reloading.com has a 'tech line and tips' icon that will then allow you to select 'advanced case forming....'. this will tell you about 'donuts' and a good deal more.
sorry, but it's really a pain to do it in a Handi' compared to a bolt gun.
ss'
it would be cheaper to throw those 6mm' dies in the trash or on the auction block. that way you'll cut your losses........