Hello Whrrr,
My brief suggestion would be to configure Hauptwerk and the DAW (Reaper) in the way described in the "Composing pieces off-line from a MIDI sequencer configured conventionally, method 2: specifying your own MIDI implementation manually ... (Hauptwerk as a ‘slave’ MIDI device)" section in the main Hauptwerk user guide (pages 273-280 in the current v7.0.0 version).
That way all MIDI goes via the DAW so you can record the MIDI within the DAW (using whatever MIDI implementation you like for Hauptwerk and other VST plug-ins). You can then tweak/edit the MIDI recording (within the DAW) until you're happy with it, then simply play it back (still from the DAW), whilst using the DAW to record the resulting audio onto a new audio track.