Get a can of black paint and wipe out the BL for the 200' before the roundabout. That will lessen the driver reluctance to let you merge early.
Seriously though. From:
http://www.azdot.gov/CCPartnerships/Roundabouts/faq.asp
"Bicycles have two choices to negotiate a Roundabout. The more avid and skilled bicyclists can merge into a traffic lane before the bike lane ends; ride close to the middle of the lane to prevent vehicles from passing and cutting the bicyclist off; enter the Roundabout after yielding to vehicles within the Roundabout; circulate the Roundabout being careful to watch for vehicles waiting to enter the Roundabout; and exit the Roundabout as a normal vehicle would do. Bicyclists not wanting to enter the Roundabout can enter the sidewalk using the ramps where the bike lane ends, and proceed around the Roundabout as a pedestrian."
Al