Support for Fisher Price Bedtime Routine System - GWM53

Is it possible? It seems that it’s controlled by some esp-like controller over Bluetooth.

That’s not an ESP like module. That’s a BLE 5.0 module.
See the datasheet here:

So my short answer will be: yes it’s likely possible to make it work with HA using the Bluetooth integration but it will require some reverse engineering of the bluetooth services this device is providing to make it work.

Is there any guide how to sniff those connections? Im pretty sure im not “technical” enough to write the addon to HA, but maybe at least i could contribute the messages which need to be coded into HA