It's worth noting, if you feel like taking a trip to your local hardware store, that they make specialty blades that fit boxcutters. Some of these are designed for laminate.
Most likely, the blades that came in your boxcutter are ok for opening boxes, plastic packagine etc. However, you can buy some designed for cutting laminate, PVC, etc. Look for blades that are thicker and may have a wavy pattern on the edge. These can be especially helpful in working with plastics. The sharper the blade, the less force needed to cut/saw. Like in the kitchen, this is safer than a dull knife that needs lots of force (that ends up slipping into your thumb - Ouch).
Similarly, rather than "homebrew" a solution, for under $10 you should be able to find a cutoff wheel for your dremel made for plastic. They often are serrated, and may have holes through the wheel that lighten in.
They are super helpful and will give a great, square cut (often without burrs).
Watch out for any sharp edges if these will be near a child's mouth. A good wash in soap and water should take care of any residual dust. Those would not be "Good Eats"