In case anybody is looking at this in the future with the same problem, thought I'd give an update. After another week of listening to this sound, which is definitely air in the cooling system, I decided to just buy the spill free funnel and burp it myself on ramps. I still had no effects in the cabin with heat or with the engine overheating, just the noise. I only used coolant that was already in the overflow tank as it was slightly above the max line. I cycled between hot and cold and did a number of 3-4k pulls for 10 seconds each. After doing this for 30-40 minutes, the bubbles were not present at idle anymore. They stayed away as long as I didn't get above 3k rpm, at which point the coolant in the funnel would rise and lower back down on the way back to idle. The bubbles would come back for about 10-15 seconds before dissipating. Eventually I called it and closed everything up as it looked like it was an issue with combustion gases getting into the system but only above a certain pressure. After filling the overflow to MAX, I still had about 20 oz of coolant left from what was previously in the system. I drove around for about 25 minutes after and didn't hear any noises, though I didn't drive hard.

This morning, I checked the coolant in the overflow and it actually finally sucked back into the radiator. The coolant was all the way down to the MIN level, which it had not been doing before. When I would check it before, it was always at slightly above MAX no matter if the engine was hot or cold. I poured the remaining 20 oz into the overflow tank and it went almost right back up to the MAX line. So it seems like the burping actually helped and removed enough air from the system to bring the vacuum back from the overflow to the radiator. The bigger issue is whether more air is going to get into the system and reverse this effect. The minor loss of coolant (starting slightly above MAX and ending slightly below MAX in total) is probably attributable to spillage. When I took the funnel off it was right at the top and slowly overflowing so a small amount was lost as it dripped over the side.