Why not just use a "drop-in" kit and a cheap 10.5" barrel?
This is probably the direction I will go with my CQBR "dedicated" .22. For a training and "fun" gun, I don't think the drop-in will be too much different, plus you have a "spare" 5.56 upper if you needed it. It could still be "dedicated," for the type of practice that I think a dedicated .22lr would be good for, I don't think the slight accuracy difference of a drop-in versus a fixed chamber insert would be too big of a deal.
~Augee