Which daughter is it that is having the birthday? One of the older, or the youngest?
The whole birthday party thing has gotten out of hand and the sad thing is, it is NOT about the kids. It's about parents either showing off or entertaining themselves because they like doing this stuff. And, yes, I get to say that because I was totally caught up in it for a while, even submitting party descriptions to a website that offers prizes. Until I noticed at the most fabulously staged party ever that my daughter wasn't having a good time. My friends were blown away. The kids - not so much.
The past few years we've scaled way back, falling back on old fashioned games or going on a hike, inviting less kids, etc., and all of a sudden I am hearing from the KIDS that I throw the best parties. At a 1/4 of the effort, and 1/3 of the cost.
Not that this will solve your most immediate issue, but I want you to keep this all in perspective. If you are in a community where this whole thing has become a competition, and it sounds like you are, the best you can do is pull out and change course. Hence, the suggestions some other posters have made about doing an outing with a few close friends, etc. While I am tempted to agree with those that say "stand up to her, keep your party, use an evite" I worry, as you do, that your daughter will end up disappointed by the attendance.
Instead, I would ask you daughter who she most wants to come, keep it to 5 or 6 kids at the most, then call those families and ask them what would be a good time for them, because it is very important to your daughter that their child, her dear friend, be there. You can also tactfully mention that you had picked X date and time, but are worried since discovering that the Z's will be having a party on the same date. But, overall, the emphasis in those calls should be on how happy it will make your daughter to have their child there. When you've got a consensus, call back with the details and get confirmation whether or not they will actually come. The mailed invites will be a reminder, but not the first notice. And it is possible that you will be able to keep your date.
You can keep your current plans with a smaller group, or change it to a destination party. Whatever you do, remember that kids enjoy the simple things. Grade school girls do great with versions of hot potato, tag, pin the tail on the donkey, and target games (especially if throwing water balloons). Preschool kids have a blast making "popcorn" by shaking balloons on top of a sheet, or tying each other up with crepe paper. Simple crafts are always fun, too.
Whatever you end up doing, remember that it is about your daughter having a nice day. The invitations aren't a very big of that, in the end. Being with a few people she cares about, and doing some things she really enjoys - those are.