However I don't quite see the Bluetooth use case - RP2040 is not really a low power chip, making it pretty hard to use for a battery-powered IoT application. You'd need a pretty giant battery pack to make it last a long time.
Nordic's nrf52 is an order of magnitude better for a typical "sleep-burst transmit-sleep" cycle, and can be suspended to <5uA current. Pico W is $6, Seeed has a $10 nrf52 MCU, or you could get a "just hook up USB and power" bare module for $5-6.