If I control BF1 with one of my units and conquer BF2 with Rumble Hotheaded, can I recycle my unit at BF1 and play a unit from my trash onto BF1 still using Rumble Hotheaded's conquer effect?

Yes, you can do this.

Because the conquer ability of Rumble, Hotheaded is a triggered ability that goes on the chain after you successfully conquer a battlefield, you can use the period after it triggers to take actions if you have priority, or simply resolve the chain item as intended.

According to the rules regarding battlefield control (Rule 187.4.c and related clarifications in the Unleashed update), you do not lose control of a battlefield if you are in the process of resolving a chain item.

Here is the sequence:

1. Conquer: You conquer BF2 with Rumble, Hotheaded. This trigger is placed on the chain.
2. Chain State: The game is now in a "closed" state because there is an item on the chain. Per Rule 187.4.c, you cannot lose control of a battlefield while there is an item on the chain.
3. Resolution: When the Rumble, Hotheaded trigger resolves, you may recycle a friendly unit (such as the one at BF1) to play a Mech from your trash.
4. Finalization: As the Mech is played to BF1, it enters the board. Since you maintained control of BF1 throughout the resolution of the chain (because the item was on the chain), BF1 remains under your control, and you successfully replace the unit at BF1 with the new Mech from your trash.

Key Rule References:

• 187.4.c: Control of battlefields cannot be lost while there are items on the chain.
• Ability Timing: Triggered abilities (like Rumble’s conquer effect) are placed on the chain as pending items after the trigger condition is met (Rule 383.4.c.2.a).
• Recycle: Per FAQ #3053, you can recycle any friendly unit from the board (your base or a battlefield) as the cost for Rumble’s ability.