If the V-grooves are exaggerated in width and depth, the simple fix is to sand the surface down until they are at the desired profile. I have a project in the works where I am doing this for a wood roof where the T&G boards are square-edge. When finished, there will be barely visible (and variable) scribe lines.
BTW, the easiest way to deal with the fuzzy grain problem with basswood is to raise the grain several times with water, sanding with progressively finer paper between applications. Then a single coat of sanding sealer, although I am not fully convinced it is necessary. This is how woodworkers deal with water-based (and even alcohol-based) finishing products.