Does Battlefield Armorer prevent gear deathknell abilities from triggering?

No — Battlefield Armorer doesn't stop a gear's Deathknell from triggering. The gear still gets Killed and sent to the trash, which is exactly the condition Deathknell checks (Rule 808.1.d), so the trigger goes on the chain as a Pending Item before the card moves. Armorer's return-to-hand is a separate zone change that happens after the death, not a replacement of it — that's the difference from something like Zhonya's Hourglass, where the kill itself is replaced and the trigger is removed from the chain (Rule 808.1.d.1).

Both abilities trigger off the same death event and you control both, so you choose the order they go on the chain (Rule 383.3.d); either way the Deathknell resolves, because it remembers the gear's info from before it left the battlefield (Rule 808.1.d.3).

Only real limit: Armorer needs the gear to be at a battlefield and friendly to you, plus the [1] — a gear that dies at your Base won't trigger it, but it still gets its Deathknell.