Did LAN issue the xONEx ticket? If not then the first thing you should do is check with whoever issued the xONEx ticket.

The issuing carrier will be able to help you more than the operating carrier.

Worse case scenario is you can always change the xONEx ticket and pay the $125 fee for changing UIO-GPS to a surface segment. Book a separate flight, and then claim the $125 change fee and cost of a new flight to your travel insurance. As far as your non-RTW ticket, cancel the remaining ticket, get a refund and then book what you need.

Option two - go a day earlier and cancel your hotel and rental car. Personally, this would be my solution as I never make non-refundable hotel or car reservations and would require much less paperwork. You could always go to UIO a day earlier, add an extra hotel night on the front to make up for the one lost on the back, and rent the car and do your thing the extra day on the front end. After all date and time changes are free on xONEx tickets. This would cost you some $ on the non-RTW ticket, unless the issuing airline's ticket agent is sympathetic and does the front end change free of charge due to the scheduling issue of your UIO-GPS flight..

Well you have a plethora of options.