Can I just the Lucian as reaction to this equipping, forcing them to still pay the equipping power cost?

Yes, you can respond to your opponent's activation of the Equip ability.

According to Rule 744.1, Equip is an activated ability. As confirmed by the chain rules, activated abilities use the chain, which means they can be responded to with [Reaction] cards or abilities.

The Interaction

1. Opponent activates Equip: Your opponent declares they are activating the Equip ability on their B.F. Sword, targeting their Lucian, Merciless. They pay the cost (the Power/Rune cost) and place the ability on the chain.
2. Chain opens: Because an ability was added to the chain, the state becomes Closed.
3. You respond: You can now play a [Reaction] card or ability (such as a kill spell or an effect that removes the target unit) to interact with the chain.
4. Resolution: If you successfully remove the Lucian, Merciless from the board before the Equip ability resolves, the Equip ability will fail to resolve because its target is no longer legal (Rule 393.1).

Important Note on Costs

It is important to understand that costs are paid when the ability is activated, not when it resolves.

• If your opponent has already paid the Power cost to put the Equip ability on the chain, they do not get those resources back if you kill the unit in response.
• The cost is considered paid the moment the ability is activated (Rule 354). You are not "forcing them to pay"—they have already paid the cost to put the ability on the chain. By responding, you are preventing the effect of the ability (the attachment) from happening, but the resources spent to activate it remain spent.