How do I snake this dual-sink configuration?
Get two bladder-style drain cleaners.



Insert one down each sink. Turn them both on simultaneously and they'll inflate until they block up the sinks, at which point they'll fill the plumbing in between the two and use pressure to push the blockage down the only remaining path.

If this drain proves to be problematic in the long-term, then you'll have to modify the plumbing. But there's no reason not to try the simplest solution first!