Robin
Realistically you can't beat stringing a wire down to your workshop and using a local Bluetooth transmitter.
That said a Google search turns up a number of range extenders which take the Bluetooth signal in and re-transmit it. Some with sticky up external aerials which, presumably, receive and transmit better than the internal type.
Theoretically best solution would be a receiver unit on one side of the wall or floor and transmitter the other with a cable between running through the major obstruction. Plenty of Bluetooth widgets with switched transmit or receive capability, I have a teeny, 1/4 matchbox size, one on my TV to run my headphones, but I don't see an official pair. I surmise a pair of the boxes I have could be made to work, just a matter of hooking up power via the USB plugs, setting one to transmit, the other to receive and an audio cable between.
Think my box was about £5 off E-Bay. Coupled with a pair of LiDL headphones cheap enough to try without being too far out of pocket if it didn't work. The official unofficial kid sister has just given me a pair of Sony WH-XB900N headphones which are very nice. Comfortable with a good sound and range even from the cheap transmitter. But for £140 I'd have expected better instructions. Large sheet in multiple languages that basically tells you nothing about how to use or set up the headphones. Major gotcha is that you set the volume on the source. No controls beyond and on-off switch and pair/reset button. As ever these days manual is on t'net.
Not Bluetooth but ages ago I bought a range extender for my DECT cordless phone that almost doubles the coverage. My phone line went into the workshop office because thats where I wanted (then) fast internet and the Siemens Gigaset base station only reached the back half of the house reliably. Repeater sits about 6 ft inside the insufficient signal line so I have signal right out to the front gate. From this I conclude the repeater principle is fine. Just a matter of set-up.
Clive