OP here with an ironic update!
First, to those who say we shouldn't invite picky friend any more, I do hear what you're saying. But she's a part of a close-knit group of friends, it would be very obvious and hurtful if she was no longer invited, and, except for this one thing, she's really a good person and I don't want to hurt her or her husband that way. So we'll keep inviting them, and hopefully I'll be able to handle food issues well in the future.
The update: we just had our Memorial Day party last weekend. Like most of the gatherings at our house, with this crowd, it was pot luck. We personally provided hamburgers, hotdogs, buns, condiments, and fixins like lettuce, tomato, onion, and I also made a tortellini pesto salad as something my own stomach can handle. Our various friends (more than 30 people showed up) brought an enormous amount of food: ribs, potato salad, red coleslaw, meatballs with various sauces, baked beans, pasta salad, chips and dips, broccoli/cauliflower salad, fruit salad, grilled veggies and hummus dip, deviled eggs, to name some of the dishes, and about 10 desserts! It really was a groaning board.
Even though it was potluck, I still was running around a lot arranging the food on the table, etc. Add to that an issue where the ribs took too long on the grill and so we ended up having to cook the hamburgs and hotdogs on the grill pan on our stove, and I was a little harried when we called folks to the buffet table. After several people served themselves, I got my own food and went to sit down, the first time in two hours that I'd sat.
A friend came from the dining room (where the buffet was) and said, "Is there any cheese for the burgers? Someone's asking."
I contemplated the cheese I had in my fridge (some plain ole block cheddar, and some special artisan cheeses that cost megabucks and would be wasted on burgers) and the fact that I would have to jump up, get the cheese out, slice it, plate it, all while my food got cold, and said, "No, sorry we're not serving cheese for the burgers."
Friend went back into the dining room to report this, and the evening proceeded well.
But I bet you all can guess who was asking for cheese for their burger?