That solution looks like it will work OK, but the drive side spacer may be thicker than necessary, because the gap between chain rings and chain stay is wider than it needs to be. Have you measured the chain line (distance from center of seat tube to the chain rings)? The "ideal" dimension for a double crank is 43.5mm to the middle of the space between chainrings. That dimension can be off quite a ways without causing problems, and you might even want to bias one way or the other depending on which cogs you use the most.

BB shims are available in 1mm thickness and cassette cog spacers can also be used here, so your 5mm could be replaced with something thinner.

About the non-DS ring being tight - I've had one bike where those threads hadn't been tapped as deep in the BB shell, so the ring tightened prematurely.