In my experience, CX have been pretty relaxed about thru-checking across multiple tickets, especially when all carriers are in OneWorld.

As for the number of flights: Back in the day, I'd had SAA thru-check me on a similarly complicated routing (CPT-JNB-ADD-KRT-AMM-MAD-MIA-GIG), across three different tickets and two different alliances. Since the number of flight segments exceeded the limits of an individual bag-tag, two were generated for each piece, with one tag continuing where the other tag left off. So technically, at least, 6 or 7 sectors shouldn't be a problem.

All this said: I'm less confident about QR being as helpful in similar circumstances, however, which is unfortunate - since I have a similar question for some upcoming travel using them as the check-in carrier:

Ticket #1: CPT-DOH-MAD
Ticket #2: MAD-MIA-GIG*
Ticket #3: GIG-CGH-SJP**

*With MIA now (again) offering ITI connections, a thru-check is especially desirable, since it obviates need to collect bags.

**Though bags must be collected in Rio to pass Brazilian customs, pre-tagged bags can more quickly be dropped off, which again, makes thru-check desirable.

Any recent experience with the QR check-in staff at CPT? Are they gems or jobsworths?