Cost is a concern but mainly trying to understand the concept. Do I have to have a controller like a Rasberry to do something like this or can it be done without?Your final choices will depend on your detailed requirements and your budget. We are aware of neither. Define these and then start searching for the devices that are the best compromise.
My local weather radio is on 162.40 mhzYou don't need a microcontroller for the radio part if you use something like this from AliExpress:
TZT Car Audio USB TF FM Radio Module Wireless Bluetooth 5V 12V MP3 WMA Decoder Board MP3 Player with Remote Control For Car - AliExpress
All you need to do is supply it with 5VDC and feed the stereo audio output to the input of your BT audio amplifier.
The weather radio requirement is much more difficult. I don't know where you could get a weather radio receiver module. All I can find is complete receivers. What ever you find, make sure it is compatible with your local area weather transmissions.