Could you splice, (or audio isolation transformer) one of the wired outputs, (say left side + -) and run a cable to one of these? and then that would go to your remote speaker (assuming mono, but look up if this is aval in stereo, if required.)

similar to u/harrybosgrandad RCA tape out, BT transmitter

https://www.amazon.com/Avantree-Wireless-Transmitter-Bluetooth-Headphones/dp/B00OPVWHRY/ref=asc_df_B00OPVWHRY/?tag=hyprod-20&linkCode=df0&hvadid=312075829581&hvpos=1o8&hvnetw=g&hvrand=6029115928859383645&hvpone=&hvptwo=&hvqmt=&hvdev=c&hvdvcmdl=&hvlocint=&hvlocphy=1013962&hvtargid=pla-521485039859&psc=1