Yes, the Arduino SAMD boards are supported by the Arduino Keyboard and Mouse library.
Nano 33 IoT is a smaler alternative to MKR 1010.
But the MKR1010 or the Nano33 IoT with original NINA firmware doesn't yet support classic Bluetooth like the HC-05 module. The ArduinoBLE library and the NINA firmware support only BLE which is different.
But there is a ...