I don't think you should rearrange. Style it out, the children will just meld together in a big crazy lump and it won't feel like anyone is missing. Next year allow more time when sending out the invitations, and remember this child will have her party near the time so check with the other mother.
This year - if you see the other mother try to have a quick relaxed chat where you acknowledge her party and wish her daughter a lovely time and without being apologetic, acknowledge that it isn't great they're on the same day - defuse that one if you can (though you have done nothing wrong, which is why you shouldn't apologise - your daughter wasn't invited, so how would you know? - but people can be prickly and weird so face into that with confident pleasantness if you can)
Rearranging will be a huge faff and will cause confusion. And it will still be really short notice so isn't great for anyone, least of all those who have accepted and put it in the calendar.
About the whole thing of "making more effort" socially. You don't have to. you are getting a lot of advice about how to fit in, like joining the PTA! - fine, if you want to join in and join the mind meld, but not all parents want to or will enjoy it, and you are not letting your dd down if you don't. If you want to be one of those women who are always at the centre of things then fine, but you don't sound like it. If your dd has particular friends she wants to invite over, then do that if you can, even if you take them to the park if your dh is sleeping - but don't feel you have to infiltrate the hive. Leave that to those who want to live that way.
your social responsibility to your dd is to teach her how to form friendships and manage acquaintances. You do that mainly by example. Running at the centre of a crowd, making big efforts to be seen and to be seen to fit in, is only one way of managing relationships and I don't think it's a great one, so if you don't bother with that why should you train your daughter that it's important? Do you want to buy into the notion that you only have functional friendships when you make a huge artificial effort to fit in with a certain group?
I think you can lead by example by being: calmly and generally outgoing (not manic about chasing a clique but being pleasant to everyone), honest, polite, welcoming when you invite people to your house, being relaxed and open when you form particular friendships and become warmer to certain people, doing this without blatantly leaving out others, respecting people and honouring them for who they are. (I am sure you already do all this.
This means: respecting the people who have already accepted and carrying on with the party as it is. Which has the advantage of being a lot less trouble!