I don't know it all acrylic that has the clear/plastic backing is the same (as in manufacturing technique), but from my experience that type of acrylic is extruded. You may want to to test your saw out on the same type of acrylic that you plan to use (which I take it, is cell-cast). If you are going to use the plastic-coated piece for testing, then leaving the plastic in place may be in order.
I have found that the protective backing of the plastic type to seperate too easily. In the past I have run blue painters tape along the perimeter of the piece to ensure that it does not come off. The downside to that is that occasionally the tape will hinder movement due to it snagging... not something one wants when making precision cuts.
In the past I have needed to go back and make cuts on pieces that I already thought were done (Girface's Testbench). I covered the surface in blue tape with a single layer wherein each strip did not over-lap (causes "catching").
Radodrill has already commented on tpi and rpm; Although I am not a professional, I have the inclination to agree with him based on my experience (or should I say mis-experiences -lol)... high tpi for sure.
I have never used a scroll saw on acrylic, so I am not qualified to comment directly, but I have used a jig saw (sabre saw). The specialised blades for acrylic come in 2 types: blades for < 1/8", and blades > 1/8".
The blades that are for < 1/8" have a double row of teeth.... since both of these tools' cutting actions are perpendicular to the piece being cut, perhaps the same logic applies: See if there is a plastic-specfic blade.
I look forward to your progress, as I have been in the market for a scroll saw (need to save some scratch... and throw more dishes away (read: donate) so there is some storage space in the kitchen).
Kudos to you for spending so much time on something for your child.
Good Daddy! ...World needs more of 'em.