Michael,
A lot of the problem with the laser cut wood is not only the "finish" that you talk about, but the fact that as the wood vaporizes, the sap, smoke and wood particles are drawn over the wood to the exhaust fans.
The recommended way to clean up a laser cutter that has become caked with residue is to use chlorine bleach. You might want to try wiping down some of the basswood with bleach to remove whatever has stuck to the top of it during either cutting or engraving.
Remember not to use too much because Basswood loves to warp with any moisture.
Rick