Well, after all of your excellent advice, here's what we did...
We let Claire buy herself a Jessie doll with her own money.
Both of the kids have piggy banks, and somehow they accumulate money - they'll get a $5 from a grandparent for Valentine's day or a $20 from another one for a birthday, or I'll have too much change in my purse so I'll dump it in their piggy banks. It's not like they have many incidental expenses. Sometimes when I know we're going shopping for stuff that will bore them (like maternity clothes) I'll let them take a purse and a little bit of their money (and sunglasses and lip gloss, of course) so they can be Big Girls with their own money... but that's really the only time they have an opportunity to spend it, and that only happens once in a great while. Generally they spend their pocket money on Polly Pockets dolls and small things like that.
Last Christmas, Claire got a FurReal cat that MG loved and coveted, so we let MG go to Walmart with some of her Christmas money from her Great-Grandma Gleason and get one for herself. I figured this was the same sort of situation. I may not always make the best decisions as a mother, but I try to be consistent.
So Grandpa Bob went to Indy and bought another Jessie doll, and brought it back, and Claire bought it from him. (The ones available at our local Walmart had plastic hair, and we wanted the one with yarn hair).
Grandpa Bob may or may not have stuck that money right back in her piggy bank, but if he did, Claire didn't see him.
I'd like to say that peace has reigned in our house since then, but unfortunately that hasn't happened. Claire is still finding things to fight over, but this attitude of hers started before the birthday festivities began. Something else is going on with her. I'd check to see if she has molars coming in, but I'm afraid I would lose a finger. If she doesn't knock it off, I'm going to make her a doctor's appointment, because she's just been acting a little bit too strangely.
I mentioned on Facebook that we had to leave Wolf Park on Sunday because she was acting like such a colossal brat. She was terrible at the zoo on Monday, too, but I just ignored her. She did fairly well on MG's birthday, until we opened presents. She's just got a hair trigger when it comes to crying and throwing tantrums, lately, and I can NOT figure out what has changed. She's getting the same amount of sleep. Her eating habits haven't changed. She isn't much more or less active. I haven't changed the way we discipline her (time outs) at all. There have been moments when she has played SO nicely with MG, and then for no reason that anyone can figure out, she loses her little mind and starts hollering about something.
If anyone has any guesses, besides teething, I'd love to hear them. She hasn't complained of being in any pain. Do kids even get molars at 3-1/2? I can't remember.