If she really is a friend, what about taking a deep breath and at least giving her a tiny bit of "benefit of the doubt" here?
Maybe she really, truly screwed up and thought your event was the Saturday and not the Sunday. If you think of her as a friend before this -- why not simply ask her? "Hey, I need to talk. I think you may have mistaken what I said about X's baby shower when you scheduled the birthday party. I said the shower was at 2 on Sunday but did you think I said Saturday? We have a conflict here and with X's baby due in four weeks it's going to be hard to reschedule her shower, and pretty much the same folks are invited to both, including X herself. What do you suggest we do here? Did you mistake the day maybe?" That leaves it in her court to express how appalled she is at getting the dates wrong if that's what happened.
If she boldly says, "Oh, I knew, I just figured that the dads would bring the kids to our party and those who want to go to the tea can leave early to make it there," you can tell her that you're upset that she knew and still arranged a direct conflict. But it's not something to explode over with her, if you have a ton of friends all in the same circle who will see you and her over and over for years to come. Handle it gracefully and they'll all know; go crazy and curse her out and they'll know that too. She is in the wrong, but if she genuinely made an error, work with her. If you're not sure it was genuine but she still claims she made an error -- still work with her. Take the high road even though you are angry.
You did say for her not to plan "specifically around me." Possibly she had no real clue that the guest lists would be identical. But you gave her the green light to plan for Sunday when you said that.
One big thing I notice -- a child's party that lasts four solid hours?! She has 11-3 on the invitations; that sounds more like an open house than a party. You could suggest she amend things for the party to be 11-1 or whatever so that it's clear folks can attend both. That may be the simplest solution. Or she may say, the idea was that it's a drop-in, open house thing and I figured folks who were going to the tea at 2 would just come on the early side and leave, and those who aren't going could come later. In other words -- she may already have thought about accommodating your time frame. But you won't know unless you ask.
Another choices is to move your event to 4-6 p.m.; or move it to the very next weekend as someone else who posted suggested. I know that a twice-delayed kid birthday party seems far, far less important that a one-time-only baby shower but her invitations are out now, and that means folks are putting this on their calendars and to me at least, the first invitation accepted is the one that has priority even if a later one is more desirable. It would be ideal for all moms involved to do the shower while all dads do the party, if you and she work together and notify everyone as a duo that that's the way to handle it. But it's complicated to do that. I'd ask baby shower mom about moving the shower by one weekend.