Just to update this thread, I removed the ceiling tiles around the area like MustangPaul suggested. That gives me enough space. Here is a picture showing the area with ceiling tiles and without. You can't see it too well, but I do gain enough inches over where I'm going to put the pin. (Ignore the kids' art work.)
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Here's another picture.
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Unfortunately, that left a number of wires hanging down.
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I found these things at Home Depot to deal with the wires:
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They work great:
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I have to admit that the ceiling still looks a bit worse than I imagined. My wife is not happy . . . even though she initially agreed. I'm going to try to make this look a bit nicer. I probably won't paint things black. I was thinking about getting some sheets to hang over the area, kind of like draping hanging tapestries from the top. Then maybe get a cheap hanging lamp from Ikea. I'll post some more pictures to show you how my game area is progressing!