Jeff,
I always use a zinc oxide barrier cream and that should not be the cause of your problems.
I suspect that the problem may be the fit of the diaper, rather than the material itself. I wear both plastic-backed and cloth-like diapers and have no chafing problems despite using zinc oxide creams. I far prefer plastic-backed diapers. However, I use the Seni brand diapers (from XP Medical) which I think are the best breathable diaper available.

IMHO the Seni diapers stretch less than any other cloth-like diaper I have tried. That can be an important factor in chafing. It is possible that, as your breathable diapers absorb urine and begins to sag, the tendency of breathable diapers to stretch exacerbates the problem. The sagging diaper, now hanging lower, rubs more and may be the cause of your chafing.
--John