Oh yeah using the built-in camera smoothing probably wouldn't work for this; I think you'd need custom logic that can be mirrored on the shader to get it to feel right. So it'd just be a bunch of extra work. Camera smoothing is also kinda hard to get right with low res pixel art games since it can end up being jerky and the feel of jumping to the next pixel is rough. There's ways around that too, obviously, but again it's more shader magic.
I do think that "spotlight" mechanic is the thing that makes this game feel unique, so finding the fun in it and tuning away as many of the problems in it as you can are probably worth whatever amount of time you're willing to spend on the game. But I think you could get 90% of the way there by something as simple as tweening the camera's position to follow the player's position and tying the spotlight position to the player's position instead of the camera's position so it tightly tracks the player sprite. I'd have to see how you put it together to be more helpful than that but I do think there's a chance you'd be able to find an easy and "good enough" solution that keeps the charm and helps the issues you want to solve. Good luck!