Hello and welcome to JustAnswer!
I watched the video and although it can be hard to tell what noises are without being there in person, this sounds exactly as you would expect a broken flexplate to sound like. If the noise is coming from the oil pan/lower/rear engine area I do believe you have a broken flexplate. Anything else in the lower end of the engine that would make noise should make a constant, timed knocking noise. A rod bearing or anything like that is going to make one knock per revolution no matter what, a flexplate often makes a more random sound like this.
The fact that it comes and goes or gets worse in gear also tells you it's very likely a flexplate issue.
Another thing you can do is try turning the a/c on and off. This changes the load on the crankshaft and will often make a broken flexplate make noise.
A broken flexplate will often give a clunk when the starter engages and when the engine comes to a stop.
If this shop doesn't believe it's the flexplate but can't tell you what the noise is I would recommend another shop. Based on the location and characteristic of the noise I believe it's the flexplate.
Thank you for replying. I also wanted to mention in the first question, (I was unable to edit it in time), that before I could get it into the shop, the car would not shift out of first gear. It barely went beyond 10mph and had to be pushed back into my driveway. At this rate, I am going to have to take it to another shop because I have been without a running vehicle for almost two months. Enough is enough. I'll stop by there and have them try it with/without the ac running. I also know that it's probably wayyy too much to ask, but would you be willing to phone the current shop and speak with the technician?
That's unfortunate, but thanks for your insight. I'll see about getting it towed as soon as I can (as towing and getting another shop look at it would be stretching resources I really don't have).
I don't know what's up this time around. When they repaired the tranny pan, I had no problems and there was only a week of down time. Since I brought it back, it's been nothing but "We'll call you back." Even after I went there in-person last week. Just another notch on the chalkboard, I suppose. I'll call around and see who's willing to even look at it. Pretty much SOL if you don't own a Ford, Toyota, Subaru, or Chrysler around here.