I think it odd to use a kid's birthday party to hold on to adult friends. People grow apart - sometimes it just happens. But to invite someone you're not close to, and who your daughter is not close to, to avoid "drama"... seems to me like manipulation, on both sides. I know the language seems strong, but - from what you say, she would have a snit if she was not invited, regardless of your current circumstances, and so that's manipulating you into doing what she wants. You're trying to manipulate her into being nice to you, by inviting her to a party. ("Bonds that Make Us Free" by C. Terry Warner is an excellent book that looks into this, in part.)
Think about what is really important here. You're celebrating your daughter's birthday. We never had birthday parties when I was a kid - we got to invite one, maybe two friends over for a sleepover, pick our favorite dinner, and choose our cake and ice cream flavors. And we didn't have to wash dishes afterwards! Parties can be stressful anyway. Perhaps just set a numerical limit on how many people can come, and then choose within those bounds. If people ask, you could say something along the lines of what Barbara H recommended, or you could say something about trying to reduce your expenses by having a smaller party, or simply that a large number of guests was just too much. Think about who is really important to your daughter, who she will have the most fun with, and then invite only them. Charity invites tend to know they are invited "only out of kindness" and not from close friendship - and they tend not to enjoy the party because they get left out socially. I know. I've been there. I've gone to parties where I only vaguely knew the person, after having taken the time to pick out a thoughtful gift, only to be lost in the mass of children and ignored by the birthday person. I don't even think the birthday person truly appreciates the gift they get from the charity invite, because they're just not that close to the person. They value the person less and hence their gift and their company.