If I used Beast Soul Swap on a Fusion monster, say, Ojama King, the Fusion would go back to the Fusion deck. Is this legal? If so, would I have to Special Summon a monster from my hand? And what if I don't have one to summon?

Yes, it is legal to use Beast Soul Swap on a Beast-Type Fusion Monster. You would Special Summon a Beast-Type monster from your hand even if the monster was sent to the Fusion Deck. If you don't have a Beast-Type monster to Special Summon, you would have to show your hand to your opponent to verify you do not have a Beast-Type monster of that same level to Special Summon.