if my opponent kills my dockside lock up can i trigger its ability?

Radiance rules aren't published yet. This uses the printed card and current rules only, and it isn't a ruling on the new mechanics.

No. Dockside Lock-Up only puts an ability on the chain in two cases: when you play it (banish a unit there) and when you hold that battlefield (kill this to play the banished card). Dying isn't a trigger condition — it has no Deathknell, so an opponent killing it triggers nothing. That includes Deploy's own reminder, "when an opponent holds here, kill this": if the opponent holds that battlefield, the gear is killed and the banished card just stays banished.

The one window worth noting is that "kill this" sits at the front of the hold ability's effect, so under the cost-in-instructions rule (383.3.b) you kill it yourself as part of finalizing that trigger — your opponent doesn't get a chance to kill it "in response" and deny you the play. Outside of that, once it's dead by anyone's effect, you get nothing back.

This is my interpretation based on Deploy's printed text and the triggered-ability rules. No official FAQ exists for this specific interaction.