You have me scratching my head in wonderment.

You say you had the cluster out so to me it means you had the top of the steering wheel cover off. It is more likely something was jiggled loose while wrestling in and out the cluster.

I would go back to the scene of the crime and look for clues! (:-)

Underneath that cover on the left side is the turn signal stalk that operates the outside dipper relay.
The power to the headlights and the relay stays outside on the fender and the turn signal/dipper stalk only grounds the relay coil to operate the relay toggle feature.

It is more common for the stalk switch to fail in operating the dipper on the newer models of the 240. At least that has been my experience anyway.

There are two switch models out there. Both are assembled with rivets but one (the newest) has only two rivets and the other has four. The four rivet models last longer and as I remember have SKF or SFK brand molded into that cover that is riveted on to the aluminum housing.

When I go parts hunting for my little "trinkets" I peek around for the older style to stash away.
I have not bothered to open these up but I suspect that there are either weaker springs or the covers strength might be affecting the alignment of the contacts inside the two rivet models.

The bulb failure sensor should not know which beams you select or when. That line of thought puzzle me too!

Do a switch-a-roo of the switches or pull the respective wires off and touch them together. I think it will make the process of elimination quicker from above and from a sitting position.
Doing the "dashboard diving" exercise is never my favorite! (:-)
Good luck
Phil