Signal Noise in Bluetooth Adapter when Using SLA Battery
It turns out that Bruce was correct when he said he suspected I have a ground loop in my circuitry. I did some research on the concept, and the way to prevent a ground loop is to have some form of isolation. In the case of audio (and many other applications) that can be accomplished through a 1:1 isolation transformer. This Wikipedia article has some good information of what an isolation transformer is.

I read this article and it was stated in the article that dial-up modems have 1:1 isolation transformers built in. I didn't have a dial-up modem but I did have a DSL modem, and inside was indeed an isolation transformer. I desoldered it and added it inline to the wires carrying the audio signal from my Bluetooth adapter to the audio amplifier and the annoying beeps/blips were completely gone. This is indeed the correct solution to the problem I was having.

The reason this works is because analog audio signals are indeed alternating current, not direct current, and the audio signal can therefor be transferred over through the transformer due to electromagnetic induction without the wires physically touching and causing a ground loop. If the audio signal was direct current (as is the case in digital audio), it would not be able to pass through the transformer (though at some point, the audio signal must be converted to analog in order to vibrate a speaker's cone and audibly hear it).

Below I've attached a photo of what the transformer looks like added into the circuit. The way it's attached is temporary. My solution was a bit more DIY, however ground loop isolators are available commercially for those interested.