Long story short, I tried to make myself a nifty TARDIS banner in Minecraft PE... After I covered ink sac patterns with lapis lazuli patterns, the result was still darker than it should've been... Notice that the MCPE in-game TARDIS has darker outlines than the other version... This might work if you cover ink sac patterns with other dyes, too, although I haven't tested that out.
Another thing I realized was that using gray dye instead of an ink sac doesn't solve the problem: the lines stay the same shade!