Cut part of the baseboard without removing it
Something great for this is an "oscillating multi tool".



They have various blades and sanding attachments, but the straight cutting blade in the picture above will probably be the best thing for cutting out a piece of molding like you are considering.

You can control the depth of cut (by hand, manually) and they can cut right next to the floor to get a complete cut.  I've used it to cut molding next to a door when installing slightly wider door casing.  I've also used it to cut the length off of door jambs when installing thicker tile (like a flush-cut saw).