Does the battlefield conquer effect have priority over other "When I conquer" effects?

No — a battlefield's conquer effect does not have priority over other "When I conquer" effects.

All "When you conquer" triggers (from units, battlefields, or other sources) trigger simultaneously when you conquer. Since they trigger at the same time, the player who controls them chooses the order they're placed on the Chain (Rule 383.3.d). They then resolve last-in-first-out as usual.

So you get to sequence them however you want — no inherent priority between a battlefield's and a unit's conquer effect.