Possible ways to add a 3rd carrier:
1) use CX for LAX-HKG and QF HKG-BNE
2) use QF LAX-AKL and LA AKL-SYD and QF SYD-BNE (very hard to get seats on this LA flight though)
3) use IB FCO-MAD and then IB MAD-ORD or AA MAD-MIA
4) add a sidetrip from FCO such as FCO-HEL-FCO to visit Helsinki
5) add a tail leg from Rome, routing FCO-ORD-DFW-JFK-VYR on CX. This adds CX as your 3rd carrier and gives you an ORD-YVR trip to use up to a year later.

Lots of other solutions, but you get the idea. The rules are clear, you must fly on 2 different Oneworld airlines to qualify for this award and there is no getting around that.