I would suggest contacting the Hyatt @ the Airport and ask them if SFO wheelchair services can pick up at the Hotel for your 'delivery' to the terminal check in area. Also I would probably ask if they would be willing to offer complimentary 4pm check out to limit your time before FIji's check in. Finally, I'd consider having an early dinner at the Hotel restaurant (never been there) to further drag out your departure from the hotel so you can arrive at Fiji's check in counter when it opens.

I can't speak to SFO wheelchair services specifically, but a couple of months back when traveling with my Mom I was surprised to learn LGA services had no problem dropping us at the fairly distant Amex lounge, picking us up about 40 minutes before departure to wheel her to an AA gate. YMMV.