RTW fares vary by airline/airlines. I thought many of them require 3 stops, not 2. You might have to stop in a 3rd city for a few hours.

The order of cities depends on your schedule. Also take into account the airline's schedule. If you are flying east, consider the 3rd city to be New York and then fly the daylight flights.

If I flew coach and was required to fly 3 cities because of airline rules, I would fly LHR-SIN SQ, SIN-SFO via either ICN or HKG on SQ, SFO-LAX, stay in LAX the minimum period, maybe a few hours, then LAX-LHR on NZ

Another problem is that some require a minimum trip length of ten days. For oneWorld's Global Explorer, for example, the rule is "For travel originating in Europe, travel on the last Intercontinental sector must commence no earlier than 10 days after commencement of the first Intercontinental sector." Other RTW fares vary in details, but generally have a similar minimum. They do this to prevent business travelers from using these fares as a less expensive alternative on things like short trans-Pacific business trips. That's just enough longer than a week with both weekends to make it hard for businesspeople with one-week trips to use them.

Amex 2 for 1??
 

Since your spouse will perhaps be travelling in employer-paid Business, have you thought of the Amex Plat 2 for 1 program?

As I understand it, your spouse needs to buy a full fare Business, but then the companion is free (or perhaps taxes only)

Good luck!