Turns out Nintendo rolled out another pretty useful feature with Switch OS version 4.0.0 but curiously kept it a secret. Some Reddit users have discovered that the update comes with support for USB devices, even wireless USB headsets like Sony's Gold Wireless headset for PlayStation and PC. You simply have to plug the device's dongle into the Switch dock, and you'll notice a new volume slider for it. The feature supports a variety of other USB headphones, so you're not limited to the PS Gold. However, it only works when the Switch is docked.
One possible workaround to make it work on handheld mode is to plug the receiver into a USB-C adapter -- it's not an elegant solution, but it's a solution nonetheless. In addition to USB support, Switch version 4.0.0 has other sweet features to offer. It also gives you a way to record 30 seconds of play and, best of all, the ability to transfer saved games and user profiles to another system.