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I'm using Godot, but I'm explicitly upscaling the render to the window size. I wanted it to render at 320x240 to keep the tiles and character small.

right - so what you could do though - is make the camera position(s) set to a specific location that goes according to screen width and then
add / subtract an "offset" variable - also according to screen size or the player. Another alternative - would be to use a LERP in order to move the camera more smoothly to the position that it needs to be.