Help -- KOA to OAK (or SJC or SFO)

Trying to book an RT award ticket to/from the Bay Area to Kona, HI the week of the 26th -- it's an out one day, back the next. I need the return leg to be non-stop because I'll have a puppy with me.

As I look at the American Airlines award chart, there are SAAver Awards in first class all that week -- out from SJC, return to OAK. When I call BA, they tell me they have no redemption space available.

Am I wrong or is this a hang up, call again issue where I just need to get a good agent? I thought if SAAver award space was available on the AA site, it meant I could book it with the Avios miles via BA.

All help welcome! It's a really cute puppy...

 

I would call again. Sometimes the BA agents don't check availability in all classes. I book OAK/SJC-HNL a lot on Alaska and I have to ask them "are you sure" and respecify pretty often.