similar, but i think the pico is smaller..
About the shield, yes its Arduino sized, but should be possible to make it smaller - the biggest parts are the wheel (you can skip that) as well as the SD Card holder which could be replacable by MicroSD..
Else.. hmm, you could see if you can get some EEPROM(s) to store waves and use some of the code from that shield.. i would recommend using I2C eeproms if possible to use less pins and easier interfacing with that small form factor... but using EEPROMS would definetly limit the size of the waves dramatically..
last possible thing i can think off.. use the shield as a template and try to wire up a mini/micro SD directly into the system using multiple small platines or omiting them where possible... the size is really a problem! G