I had the same problem in a hive last fall. Hive-top feeder, not division board but the same experience. That hive went into fall weak and subsequently crashed, supporting your hypothesis that it's mostly a problem for hives that are already weak.
I didn't try bleach, but vinegar did nothing at all to deter them.
I have problems with SHB again this year, though I have not yet seen them in any of the feeders. Mechanical traps (BeetleBlaster, et al) are not keeping them under control anymore. (And the diatomaceous earth in a CD case trap was a complete flop.)
For context, I am in northeast Ohio but there is a compost heap within about 10 feet of the hives.
Mike Rossander
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