dBpoweramp offers secure ripping and metadata tags from several online sources. One of more of those sources provides separate composer tag values.

J. River Media Center 15 is an all-in-one program with secure ripping, wide support for the tags you need and good WASAPI and ASIO support for bit-perfect audio output. Its online tag database does not have separate composer tag values.

JRMC has features that make it faster for me to hand enter tag values for classical music than to modify the values that dBpoweramp provides.

Here are some JRMC screenshots showing how I use tags for classical music.

http://naturelover.smugmug.com/Other/JRiverscreenshots/9497548_8tZYK#637838777_Ybocp

http://naturelover.smugmug.com/Other/misc/8915115_aSrUm#591669097_CuBXr

Neither program is free but they are worth what they cost.