I think trying to play the track labelled Cliff Burton (Bass) and then tapping the track "Lead Cliff 1" and ignore "lead cliff 2". You will need to learn to finger open strings with your fretting hand, and it seems like such a pain to do!
To be honest I can never hear these famed "bass solo's" on the MoP recording, even on good stereo's. To my ears it sounds more like the guitars thicken up.
Also, my "official" Metallica transcriptions books (notation and tab published by Cherry Inc.) have always been very accurate and the MoP book does not list a 2nd bass track for Orion.
The same author does list a second bass track on the intro to For Whom The Bell Tolls (Ride the Lighting book), so when it's required, it seems they do list extra recorded bass tracks. Just not for Orion
Either way have fun playing it it's an awesome song.