I found a discount on a USB-A or USB-C transmitter that can enable you to connect Bluetooth headphones to multiple device types. Seems a good little option where the device is lacking bluetooth capability. From an Australian warehouse so delivery time is shorter and GST covered.
COUPON CODE is : BGBWBL2AU
Model: BW-BL1
bluetooth Version: V5.0+EDR
Supporting Protocols: A2DP, AVRCP, AFH, HFP, HSP
Input Port: USB TYPE C / USB A(transfer line)
Power Supply: Console or power adapter
Support Dual stream: Support (one transmitter is connected to two bluetooth speakers or headphones at the same time, and one codec is used)
Bluetooth Distance: ﹥10M
Compatibility: Switch, PS4, TV, Laptop, PC, Projector
Size: 30mm45mm10mm
Package Includes:
1 * bluetooth V5.0 Transmitter
1 * USB-A Adapter
Just to let you know if you have a thick protective case on your switch like Skull and Co grip it wont slot into the USB C port