I don't know if this is possible in Logic, but Sibelius has excellent part management capabilities, so you'd likely be able to find a good way to accomplish the split you want before exporting to MIDI. There might be an easier, more elegant way of doing this, but off the top of my head I can think of this:
1. After your score is finished, save as to create a new version that you're going to mess up a little. Call it "song name for export" or something like that.
2. Add a new staff for each voice you're going to extract from an existing staff. For example, if you are writing for chorus and you have one staff for male voices and one staff for female voices and you want to create a separate MIDI track for Soprano, Alto, Tenor, and Bass, you would add two staves.
3. Select all the measures of one of the staves you want to split, and use the filter to select only the notes for one of the voices in that staff.
4. Cut those filtered notes and paste them into your newly created staff.
5. Once you've done this for all staves with multiple voices, your score is ready to export.
If the bulk of what you're doing is making piano reductions into larger arrangements, look into Sibelius' Explode function.
If there is a quick and easy way to accomplish this task in Logic, I'd love to hear it, too. But I think this kind of process is likely better handled by Sibelius.