So the thing that sticks out to me throughout this conversation is that you say the L3 and L4 coaches/gymnasts are good at this (current) gym. What I cannot figure out is how they GET a good L3 team if they aren't adequately coaching them in L2. Are the L3/4 coaches picking kids just based on potential from the L2 team? If the L2's aren't being trained properly with an eye on higher level progressions and skills, where is the training coming in? HOW are the L3's getting good, only after moving up? We don't do L2 at all but instead have a preteam/developmental group. It is the job of those coaches to make the basics, form, strength and skill progressions good enough for L3.
Playing devils advocate for a moment (bear with me)... is it possible there is something, anything, else that could be the culprit? Is your DD focused and working hard during practice? Does she actively try to use corrections etc? Has she been dealing with fears or blocks that may make them think it would be in her best interest to take things slower?
It is positive that another gym would put her on their L3 team, it shows she at least has the potential to get there. However, I would truly think about if there are any other circumstances that could come into play that your "mom goggles" may be missing...
If there isn't anything at all, I would seriously consider moving gyms. I would go over everything with a fine tooth comb before though because you keep saying the other coaches are great and I would hate for a kid to miss out on what would be a great gym for her simply because a parent feels she is inappropriately placed in levels...
Do realize that I am speaking from a coaches standpoint here. I have seen girls leave our program for that very reason, go to our main competition because they were promised the level they wanted (which they were not getting at our gym, for various reasons) and those gymnasts (without a single exception!!) have ended up either repeating that level (thus ending up in the same exact spot they would have been with us a year later) or failing altogether. Not saying that is the only outcome, obviously there are many success stories, but that has been my personal experience with our gym over the past 5yrs being involved with team. Generally, if we are holding a kid back, there is a really good reason and it has proven to be correct with every single one of the kids who moved.
The opposite could happen, if the coaches at her current gym aren't as good as you believe and they are for whatever reason "skipping over" her and not seeing her true potential, then absolutely you need to move gyms and go where her coaches will be more invested in her.
A larger gym may have less "invested" in each kid and allow more kids on their team to begin with, with the idea of coaching to the top and letting the bottoms repeat.
I know this post is all over the place, I am just trying to allow you to really take an objective look at it. It is really easy as a parent to just jump on to the "well if she isn't moved up here you need to change gyms!" bandwagon, nobody really wants to say that hey, maybe there IS a reason she is being held back. Us coaches do not WANT to hold a kid back. What we DO want (most of us anyways!) is to place a kid where she is going to have personal success and fit in, so that she enjoys gymnastics and keep progressing at a steady rate without being too overwhelmed or frustrated. Maybe for your kid, that is L3, maybe it isn't at this point.
Best of luck, it is such a hard thing to make these decisions for our kids sometimes!!