You could try the following..
cai-lhr-bom
bom-bkk-sin (though this would put you over the segment limit.. maybe just bom-bkk)
sin-hkg-pek
pek-hkg-yvr-dfw-ccs
* Buy a separate ccs-clt R/T (though not looking cheap at the moment, LIM or UIO might even be cheaper)
ccs-scl-puq-scl-mia-iad
iad-lhr-mxp
mxp-lhr-cai

* Or this could be done in reverse..
pek-hkg-yvr-dfw-clt
clt-mia-lim-scl-puq-scl-ccs
*CCS-IAD return, then you could fly direct CCS-MAD-MXP

From the oneworld validator:
1: CAI-LHR: 1st of 4 (Europe)
2: LHR-BOM: intercontinental
3: BOM-BKK: 1st of 4 (Asia)
4: SIN-HKG: 2nd of 4 (Asia)
5: HKG-PEK: 3rd of 4 (Asia)
6: PEK-HKG: 4th of 4 (Asia)
7: HKG-YVR: intercontinental
8: YVR-DFW: 1st of 6 (North America)
9: DFW-CCS: intercontinental
10: CCS-SCL: 1st of 4 (South America)
11: SCL-PUQ: 2nd of 4 (South America)
12: PUQ-SCL: 3rd of 4 (South America)
13: SCL-JFK: intercontinental
14: JFK-DCA: 2nd of 6 (North America)
15: IAD-LHR: intercontinental
16: LHR-MXP: 2nd of 4 (Europe)
17: MXP-LHR: 3rd of 4 (Europe)
18: LHR-CAI: 4th of 4 (Europe)

Congratulations!! Valid itinerary!!! - (41347 miles)

Though pricewise it may end up being a wash.. buying a LONE3 + South America ticket ($2000 + $1000) might be the same as a LONE4 + add-on ($2400+$600).