I agree with scroca. There should then be a syncronized crosswalk light accross that island where the cars make the last right turn. There is NO other possible simple and safe solution. I would say it's worse than a highway merge lane.

As is, they really only moved the double turn lane to infront of the light. Better but you still must cross 2 right turn lanes one at a time.
I would head for the left of the must turn lane, close to the lines, before the light (but not very far ahead). Don't stay in the middle of the lane. If the light is red while riding up the shoulder, cut behind the first car if possible. Else, make eye contact with the front driver.
Crossing the intersection, I would then head for the CURB until figuring out who is turning, and or wait for the red light. I would probably keep the lane by the curb to near the end of that next block and have a cars must yeild sign. Even if you knew the first 3 cars were going straight from the light, it's still a bit far to the safety of the inside bike lane.

Another possible design would be a bike box in front of the second lane, before the light. Extend the left side bike lane closer behind the light. Have controls to stop right turns the first 2/3 of the red light phase for bikes to enter the box (or go totally across with peds). Have a bike only pre-start for 2 secs to get to the bike lane on the other side. Allow NO other thru bike movements. Good luck with that. Otherwise right turns OK either bike or car. Then what about after hours ...

PS : No lights/sirens will make a speck of additional difference here.