You can legally straddle a shoulder stripe, so make sure it is a designated bike lane and not a wide shoulder. Actually, you can ride just to the left of a shoulder stripe no matter how wide the shoulder is - the 'stay to the right' rule applies to bicycles operated on a roadway and the definition of roadway excludes the shoulder.
If you are going at the normal speed of traffic, straddling the bike lane stripe is probably legal - you'd have to find a section of the vehicle code prohibiting that, and I don't know of one. If you find one, please let me know. Otherwise it would depend on whether the stripe is part of the bike lane or not. Normally if it was a bike lane, I'd stay just to the right of the stripe, unless one of the exceptions listed above applied, in which case I'd do whatever minimized the risk of an accident, letting faster traffic pbutt easily when safe.
Also, if the street has driveways and cross streets mile after mile, so you are always approaching a place where a right turn is permitted, a bicyclist (in California) does not have to use a bike lane. Riding on the strip puts the cyclist farther to the right than legally necessary, so I don't see how riding on the stripe would be a problem in this particular case.
-- My real name backwards: nemuaZ lliB