Like most things chicken, you may just have to try it and see how it works out for you.
White plymouth rock to Breese is also something to consider. There are a few threads on here of folks raising straight Breese for meat, and they can really bulk up over the generations if you select your breeding stock correctly and cull appropriately.
The commercial CX come from 4 parent flocks, and they use dwarfism in one or more of the flocks to keep birds that will be able to reproduce. The terminal cross chicks that are sold commercially can't - by the time they're old enough, they're too big. Sometimes folks can keep a CX hen and cross her with something and get a few eggs to improve their meat flock, but the CX hens usually die before they're 2 yrs old regardless of how well you keep them.
I tried the White Rangers from Freedom Ranger hatchery, which are very, very similar to CX but with better livability, and was very happy with their ability to give meat and eggs in their first year. I'm into my second year with my remaining hen, and she has not successfully laid yet this year - she's giving me soft shell eggs, but I don't know if she did that last year also. I tried to keep back both my hens, but one died. She may have been the one that was laying.