I would either:
1. Choose another route (My first choice. LA has lots of streets; surely there's a way around this stretch of road?);

or

2. Go ahead and ride on the road. It's a designated bike route, so there may be a bit of room even with parked cars, and, more importantly, there are traffic lights every few blocks; the cars really can't build up much more speed than a bike before they too have to stop at a light, and it's only 1.5 miles long. If a few motorists get irate because they get to the next red light a few seconds later than they otherwise would, so be it. They're probably angry about a host of other things that have nothing to do with a guy on a bike anyway.