A Godot shader which reads the lowest four bits from the input's blue channel and uses them to pick a substitute colour from a palette texture.

palette_remap.gdshader

// Reads the lowest four bits from the input's blue channel and uses them to
// pick a substitute colour from a palette texture.
//
// The palette texture should be 16 pixels wide and 1 pixel high, with the
// colours arranged with index 0 at the left and index 15 at the right.

shader_type canvas_item;
render_mode blend_disabled, unshaded, skip_vertex_transform;

uniform sampler2D palette;

void fragment() {
	float blue = textureLod(SCREEN_TEXTURE, SCREEN_UV, 0.0).b;
	int index = min(int(blue * 256.0), 255) & 0xF;
	vec3 colour = textureLod(palette, vec2(float(index) / 16.0, 0.0), 0.0).rgb;
	COLOR.rgb = colour;
}