I looked into this and I'm confused by your problem,
The shadowed sprite shader is affected by lighting and ambient light, so switching color space will affect the final rendered color (this also happens with regular geometry using the standard unity material). The shadowed shader is basically just the old default Lambert-shaded standard shader, with a shadow pass.
Linear space isn't reinterpreting the canvas colour. What you're seeing as darker colours is just the lighting from your scene affecting the sprite, however you have it set up. The colors look different when you switch color space because the colors and intensities of the lights in your scene remain the same and should be adjusted so the scene looks the way you want it to in the color space you'll use.
If you don't want the colors of the sprites you drew to be affected at all, you can use an unlit shadow-casting shader (not included in doodle studio but lmk if you need one).
Worth noting is, the UI of doodle studio seems to change in linear vs gamma space (which seems to be a unity bug), but the colors in the texture you're drawing remain the same. So you shouldn't need to switch spaces to draw. You should be able to stick to one color space.
Color spaces are confusing! Let me know if this clear things up