I would not worry too much about Lithium-Ion safety. We use single-cell Lithium-Ion batteries in a lot of our products and projects and, just like the LilyPad MP3, we include a LiPo connector and charging circuit since Lithium-Ion batteries are a great option for a lot of power in a small package. That said, there are some minor risks when using Lithium-Ion batteries you will want to consider when using it in a project. I’ll outline those below.
There are a few ways a LiPo cell can fail catastrophically but for the most part, they are completely safe. The two primary ways we have observed (and done in “controlled” environments) are physically piercing or cutting the cell and creating a hard short. Physically damaging the cell is what usually causes the catastrophic, firey failures you may be thinking of but with the proper precautions, this is a very rare failure. Creating a short is the more common failure but so long as you are careful not to short the positive and negative terminals of the battery there will be no issue. If a short is created, sometimes the battery is fine but a short can damage the chemical structure of the cell which usually results in a “ballooning” or swelling battery. If you notice this, you want to remove the battery from your project follow the proper disposal process for them.
Now, for your project specifically, so long as the battery is connected properly and protected from any cutting/piercing, you should have no issue using one but if you are concerned about that or if your project will not be able to contain the battery effectively, a 4xAA battery pack may be a better option. I would not connect it directly to the JST connector as that circuit is set up to accept and charge a single-cell Lithium-Ion battery. Instead, I would recommend connecting the 4xAA’s power and ground to the “VIN” and “GND” pins labeled on the board. The [User’s Guide will have more information on powering the board and a whole lot more about the LilyPad MP3 to get you started.