If I use Rumble Hotheaded's ability to recycle a mech unit at a battlefield, would I be able to play the unit from trash to the same battlefield that now has no units?

Yes, you can do this.

According to RiftJudge FAQ #9975 and #11123, when you conquer a battlefield with Rumble, Hotheaded, the ability is a triggered ability that goes on the chain. Because you are in the process of resolving a chain item, you maintain control of the battlefield even if you recycle the last unit there.

Here is the step-by-step sequence based on the established rules:

1. Trigger: After you conquer the battlefield with Rumble, his conquer ability is placed on the chain as a pending item.
2. Chain State: The game is now in a "closed" state. Per Rule 187.4.c, you cannot lose control of a battlefield while there is an item on the chain, even if the battlefield currently has no units.
3. Resolution: When the Rumble trigger resolves, you recycle the friendly Mech unit from the battlefield (which is a valid location to recycle from, per FAQ #3053).
4. Finalization: As the new Mech is played from your trash onto that same battlefield, it enters the board. Because you maintained control of the battlefield throughout the resolution of the chain, it remains a valid location to play the new Mech.

This interaction is specifically confirmed in RiftJudge FAQ #9975 and #11123.