> On one hand, the fact that the same sound can be named A# and Bb may be puzzling for a kid

I think that, given the toy is (currently) diatonic, and doesn't really have any ability to visualize the chromatic scale (like a piano keyboard does), using the formally correct note names is more intuitive. That way, only the accidentals change when you change modes ("when I change it to C minor, the B becomes a Bb"). This naturally teaches a simple and correct mental model: "the slider chooses a letter and pushing the orange knob makes letters flat or sharp".

If you only ever use sharps instead of sticking to the correct notation, then the notes change inconsistently between different keys ("changing from C major to C minor turns the B into an A#, but changing from C# major to C# minor changes the F into an E"). This is incomprehensible unless you've already memorized the piano keyboard layout.

The OP's choice of restricting to the diatonic scale seems sensible to me -- it helps the kid learn the vocabulary of Western music (if that's your goal!) and it benefits the parents as well by making it hard to create something that sounds bad.

Why stop at 12 semitones? That's so square and limiting, man...

https://en.wikipedia.org/wiki/31_equal_temperament