If I play Beast Below with no other unit on board, do I have to return it to my hand due to its Play effect?

No, you do not have to return Beast Below to your hand.

According to the RiftJudge FAQ #9370 and FAQ #466, the rules work as follows:

1. Playing the Unit: You are allowed to play Beast Below even if you do not control another friendly unit.
2. Triggered Ability Requirement: Its "When you play me" ability requires you to target another friendly unit and an enemy unit. Because valid targets must exist for all required targets to place an object on the chain, you simply do not put the triggered ability on the chain if you do not control another friendly unit.
3. Result: The unit enters the battlefield normally, and the ability—which would have returned units to their owners' hands—never triggers or enters the chain.

Because the ability is not on the chain, it has no effect, and your unit remains on the battlefield.