Update: So I experimented with a few things. First, I tried resyncing the gamepad to my Wii U (for the billionth time). Long story short: didn't change anything. I then tried unplugging and replugging: still didn't do anything. As a last ditch effort, I thought I would move the console to the original place I had it and have it standing up. This fixed the issue. There's still unique circumstances in which the connection drags a bit, but it's not nearly as common now and I can even cover the gamepad with a pillow or blanket and it still connects to the console, unlike before.
This seems to be an issue with the architecture and placement of things in my room rather than an issue with the hardware. Regardless, I'm glad it's resolved and happy I didn't have to fork over some cash for it to be normal again.