Here’s what I did:
Thread all three screws into the housing loosely
Push the housing to the lip on the plate and at the same time tighten the thumb screw all the way down
Once the housing is tight up against the lip, tighten the two Phillips screws
I “screwed around” with this for quite a bit and I totally feel your frustration. The same issue happens with the strobe. The thumb screw is the single point that holds it in place and it wiggles. I’m half tempted to spray some Great stuff into the gap between the two pieces.
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So you switched the strobe over to the other side and flipped the tray's lip to the front?