Turn signals are not (normally) part of the Bulb Failure Sensor circuit. Check the ground wiring in your left rear taillight assembly. The current from the turn signal bulb is finding an alternate path to ground via a backfeed through the taillight circuit on that side. This triggers the BFS, as there is no balancing current on the right taillight wiring. It won't do this if the ground wiring and connectors are clean, tight, and have 'no' resistance.

Just a guess, but maybe when the headlights are on there is such a large amount of balanced current on both left and right circuits that the small difference caused by the bad grounding of the left side is insufficient to trigger the BFS.
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Bob (son's 81-244GL B21F, dtr's 83-244DL B23F, 'my' 94-944 B230FD; plus grocery-getter Dodge minivan, hobbycar 77 MGB, and numerous old motorcycles)