Should the Arc be able to transmit the sound to the Port which, in turn, sends the sound to the connected Bluetooth transmitter which outputs the sound to wireless Bluetooth headphones?

You should be able to play music through the headphones while they are connected to PORT.

The Bluetooth headphones pair with the Bluetooth transmitter but no sound comes out of the headphones.

Got it to work!  I had the Fiio set to the wrong output ‍♂️

You mean the wrong input? Easily done, as the BTA30 Pro defaults to USB initially.

Thank you to everyone who provided comments and suggestions!

I had the BTA output set to optical

i originally had the BTA30 input set for ”OPT” rather than what it should have been, “COX”.  Since both inputs are optical in signals, I confused them.

I do experience some drop outs in the audio stream but that may just to be expected with Bluetooth.  I don't know if there is any good way to improve it without physically moving the BTA30’s location.

Thank you again for your help and comments.

There seem to be a lot of people who think that the world is centered around Bluetooth -- which they find to be perfect. I find it to be frustrating at best. I use a SONY WH-1000XM4 primarily for its noise canceling feature. I’m not experiencing any dropouts with Windows 10 or my iPad, but an A/V receiver with a built-in Bluetooth capability briefly mutes every few seconds, depending on the service and how I’m using the service on the receiver. The transmitting Apps and devices are constantly fighting with each other. Windows 10 seems to be more reliable than my iPad in this respect. The A/V receiver’s Bluetooth capability is only of academic interest to me.

Make sure the transmitter is well away from any device that uses 2.4GHz WiFi. Also, the choice of codec and quality (if using LDAC) are likely to be factors. Finally, don’t expect too much from the ‘long range’ description. Walls and floors will attenuate the signal significantly.