For my first attempt, I hadn't actually caught Reshiram, Zekrom, or Kyurem. I genned them into my game using PkHex and Cyber Save Editor 2. I made sure I genned them correctly, and even tested as much by trading online. It was successful. I then soared to Gnarled Den, and picked up the DNA Splicers to fuse Kyurem with Reshiram. I got "It cannot be fused with that Pokemon". I then tried fusing it with Zekrom and got the same message. I also fused in the correct order, so I first selected the DNA Splicers, then used it on Kyurem, then either Reshiram or Zekrom. At this point, I thought it might have been the fact that I was using genned Pokemon, so I actually caught Zekrom and Kyurem, then tried fusing them together. Still got the message.
I made sure I was fusing correctly, and I definitely chose Kyurem first. I still got "It cannot be fused with that Pokemon".
Thanks for replying guys, but I actually just fixed this problem, and I think I've narrowed it down to the root issue.
A few weeks back or so I tried the O-Power cheat code within PkHex, and I was initially weary about modifying my game beyond injecting Pokemon/Items, but I figured it was safe. When I started having issues yesterday, I realized it definitely wasn't normal. I thought back to what might have messed up my game/save, and I recalled adding the cheat code. Luckily, I had an old save I backed up from about a month ago, before I used PkHex and obviously the cheat code, so I restored to that point. It fixed the issue. I used the same genned Pokemon, picked up the DNA Splicers, and fusing immediately worked. Obviously, there's no way I know for certain that the O-Power cheat code caused this issue, but it's honestly the only thing that caught my attention.
Regardless, the issue is now fixed and luckily my old save was complete, and after backing up and restoring my new save's PC Data, I didn't really lose anything other than about 100 hours of playtime in my stats. Like I said, I don't know if the cheat code caused this issue, but I would use them with caution and back up your save before using them.
Thanks again for the suggestions, guys!