I don’t think you’ll get any ready- made bluetooth module running at both AC and DC. Rather, you may have to develope your own system using a Bluetooth module plus some accessories like ac-dc adapters, battery chargers, relay modules etc…
I was thinking some form of a step down or voltage regulator to convert the AC when plugged in. This could putout the 3 to 5 volt needed for the board.
At first, you’ll need to convert the AC voltage to DC. Then you may need to step that down. If you use a DC adapter, you’ll get 5V DC output directly from 220V /110V AC. If your bluetooth module runs at 5V power supply, you can use the 5V DC. If your module runs at 3.3V, you’ll need to use a buck module to convert the voltage from 5V to 3.3V.