Can audio go to both the speakers and a bluetooth headset simultaneously

Here's a good enough working solution on windows 10 that does not require any additional software installation:

1. Right click on the sound icon and choose "Recording devices"
2. There should be a device named "Stereo Mix" in the list. However, if you can't see it, right click somewhere on the list (on a device or on the free space around it) and choose "Show Disabled Devices"
3. Right click on "Stereo Mix" and Enable the device.
4. Right click on "Stereo Mix" again and go to Properties
5. Under "Listen" tab, check the box that reads "Listen to this device"
6. Finally, on the "Playback through this device" drop down, choose your bluetooth headphones device.

Now set the Speakers as the default playing device, and the sound can be heard from both the speakers and the bluetooth headphones. Adjust volume of the devices in volume mixer.

I should also note that the quality of the sound in the headphones drop a bit, and a tiny bit of delay is also introduced (i guess it wouldn't work if you're a DJ)

See if Virtual Audio Cable will work for you. You can basically create a virtual audio device which will send sound to both the speaker & BT headset.

I like to use the open source program Audio Router

https://github.com/audiorouterdev/audio-router