I am not a former gymmie, just a mom and my experience is purely from my youngest doing gymnastics. So not really knowing how to tell how my daughter would be progressing, (other than comparing her to others and watching her grow), I would probably go ahead and take her to the other gym. You know her best and if she seems bored now another year without progression is not going to be fun.
It's really a shame that these two gyms have a "feud", and I do agree that they are in it to make money, and that the rumors will fly, but it's all good experience within the competitive world of gymnastics and this is pretty mild stuff. It's not your concern what kind of rivalry these two gyms have. You are a paying customer, and if you were told one thing and that got changed, you have the right to know why, and to look at alternatives. Your concern is obviously that your gymmie doesn't get bored or upset from being let down. She is only at level 3 which at most gyms isn't even a team, but she should be enjoying herself at this point while she is progressing and learning. It was wrong to tell you she was going to be moved up, only to change their minds. If they have been coaching her, they already know what her personality is, and that she may need to focus, so that would be a lame excuse for me.
Be realistic though, and talk to the coaches from both gyms to decide on what is right for her. As mentioned by NotAMom, it's hard sometimes to make the right decision. My gymmie got moved up last year prior to the competition season because she got a couple of skills for the next level. The result=we had a rough season, she didn't score well, got frustrated and wanted to quit. So we are repeating the same level for the coming season, even though most of her teamates are moving up. She can still uptrain, and hopefully have a successful competition season, and get back to feeling good about where she is at.