Hey, sorry for the late reply...
In the end, the problem goes much deeper than messed up colours, it would seem. It turns out the whole sprite is actually the original one, but using the palette with my custom colours. This means that when the protagonist slides from right to left in the very beginning of this animation, the game actually uses the ORIGINAL SPRITE. Since my custom sprite isn't that different (in terms of pixel placement), I couldn't even tell that it used the wrong sprite altogether. So this is a completely different issue.
As for the colours, I found a fix, because it turns out it all has to do with the order of the colours on my palette. I tried changing the order, and surely enough, the sprite used different colours in different areas. Then, all I had to do was find out in which order the sprite picks up the colours, and place the right colours in the right place on the palette. This is satisfying enough for me because my sprite is not too different to the original one aside from colour, but if for instance one was to replace their sprite with a COMPLETELY different-looking protagonist, then the issue would literally have nothing to do with colour and all to do with sprite usage.
I think my specific case can rest for now, however, to anyone who has some insight on this new information, you are welcome to continue the thread to help out people who actually need the sprite to change. I'm actually curious, tbh.
Thank you to the person who helped me investigate this issue further, FAST6191.