Thanks for the tips, eishiya.
You too, strawberry. I actually reinstalled Aseprite because you can view an animation as a tile. It might be my best bet for this, I guess. Still, it'd be even better if I could just animate the entire tileset directly...why the hell does Pyxel Edit use that weird shitty "view all frames at once!" thing instead of just using normal frames? It'd be so much better if it just let us flip through frames in tilesets, hahah! Ah well.
If anyone else has any recommendations, I'd be glad to hear them. For now, I'll just be copy-pasting tiles out of Pyxel Edit into Aseprite to animate them. Now my workflow will have three pixel art programs in it! Yay! :'D