I have many tens of thousands of Amtrak train miles behind me, and while many hours delay are commonplace on long distance trains, the Surfliners are more local, and I would not expect to have to overnight in LA on that connection, myself.
The easiest option is to buy a couple of new tickets from Santa Barbara to Los Angeles for an earlier train, and just keep the original booking for the Southwest Chief.
I believe Amtrak booking staff are a "mixed bunch", some are very knowledgeable, some not at all.
I believe the term to use is that you want to MODIFY an existing booking, rather than cancel and rebook, might be worth another shot on the phone, ask for a supervisor too, if no luck...
Edit: I don't want to give you bad advice, but I am wondering now if Surfliners are "reserved seats only", you may be able to use your existing tickets on an earlier Surfliner. Worth a look?