Alright, this is awesome information and I just want to say thank you before I continue. I tried to answer all the questions asked.
Guitar is a Hamer Artist with a set neck. The bridge is an All Parts bridge that I got from the person at KMC Music (they bought out Hamer in '08 I think) who's handling warranty stuff. This is the third bridge -- presumably, the first was the same as this one. The second was a Nashville Tune-O-Matic. I'm not sure how to answer the question about the saddles being filed; do you mean were they filed in the slot? Or filed across the top to make the entire thing lower? They were filed in the slot, I know that much. I tried filing some lead and lubing the nut but that didn't make a difference. Frets aren't loose. Pretty sure the sound is coming from the saddle; I've spent the last year, roughly, holding my ear to the guitar and listening for the sound. Searching for the sound, maybe?
About raising the action, interestingly when I raise the action the sound gets worse. That feels like it should be an indicator of something, but I don't know what. About the bridge saddle being level, how would you recommend I check that? I wouldn't even know where to begin.
Unfortunately, though I have a normal straight edge, I don't have a slotted straight edge. A quick Google search revealed that they're not hard to make by oneself. Still, I have no radius gauge or string height gauge. My mechanical engineer friend suggested I use a set of feeler gauges (which are, of course, much cheaper than the string height gauges they sell online) and I think that'll work ok, though correct me if you think I'm wrong, but that leaves me with the radius gauge. Again, a little research shows there's a PDF printout that can serve as a replacement, but that's the only one I'm not really confident about. Paper cutout? Would that work?
I'm in Miami, for those wondering about my geographical proximity to knowledgeable repairmen in their areas. Always wanted to visit Wyoming, though... lol.
Here are some pictures. If there are any angles or other pictures y'all would find helpful, don't hesitate to ask. Based on what you were telling me, here's my best guess at what angles and closeups would help. Nothing like taking macro shots to make me want to yank out the dust rag...