The short answer is that they should both group as well if they are equal guns. Now for the reality, you are on the other end of the equation and you already know that the 223 will have a little jump, so you are probably going to try to compensate for it and open the group. The tightest groups that I have ever seen were shot with 17hmr rifles @ 100yds. This only makes sense since the rimfires recoil is so small that the cross hair in the scope never moves!
Using two bags, you will need to adjust them forwards and backwards until you know where the bags need to be on that rifle to shoots the best. For shots from 0-250yds I use my right hand on the trigger and my left on the back bag. After 250yds I use my left hand to steady the rifle on the front bag ( I can see my heart beat in the scope ). Play around with it and you will find what works for you. Good luck,
Qaz