I'm sorry about your loss.

I found a round trip fare on SQ from LAX to ICN (via SFO with an intra-state connection on UA) for $809. This fare is in B class which earns full miles. No advance purchase is required.

Here's a sample itinerary:

Thu, Oct 7

United Airlines 926
Depart: 11:10am Los Angeles, CA (LAX)
Arrive: 12:28pm San Francisco, CA (SFO)

Singapore Airlines 15
Depart: 2:10pm San Francisco, CA (SFO)
Arrive: 6:45pm Seoul, Korea, Rep. of (ICN)


Mon, Oct 11

Singapore Airlines 16
Depart: 5:30pm Seoul, Korea, Rep. of (ICN)
Arrive: 12:20pm San Francisco, CA (SFO)

United Airlines 280
Depart: 2:30pm San Francisco, CA (SFO)
Arrive: 3:50pm Los Angeles, CA (LAX)

I would buy this ticket from SQ directly to ensure that it's in B class so that you can get full miles in DL (on the international segments) or UA. The B class fare is refundable minus $75 fee and changes are permitted, also for $75.

RichardInSF has a good point: Award availability opens up often at the last minute. I would check back every day up until day of departure and see if you can snag a J or F class seat.