Cutting Ceiling Molding Wihtout Removing It?
One approach to this problem would be to use an oscillating tool, of which there are many brands available. Different blade options are provided with the tool, including a toothsome flat blade with a square face.

Home Depot is one source of these tools and blades:



It won't necessarily be fast, but it will be a clean cut (depending on operator skill). If you need a cheap solution, Harbor Freight is likely to be a good source. They have a twenty dollar (US) tool that does not specify that it comes with the cutter, but those are seven bucks each in the same location.