One simple way to do this with the anchor you have (assuming you get it back):
Fishing - shackle the end of the chain to the trip point. Using cotton or hemp cord (or zip tie), secure the chain to the pulling end of the anchor. This works best with an anchor without movable flukes like a Bruce or your Delta, where the chain can be stretched tight from the trip point to the holding end. Then the cord only has to hold against lateral forces, not the main pull.
If anchor is stuck fast to the bottom, secure the rode up tight and then power upstream to break the cord. At least that is the theory. This will require some experimenting to get the right cord and the number of turns through the chain link that will hold when you want it to but break when it has to. You have an advantage being a diver, you can test and observe the anchor with someone topside to run the less interesting part of the testing.
Diving - just move the end shackle from the trip point to the holding end of the anchor leaving the cord out of the system.
I was thinking that a fishing anchor might be worth looking at but they would probably not break out if stuck in a rock reef since the slider does not go all the way to the retrieval point in most anchors:Example
And I am not sure that the shackle would reliably slide up the stock reliably.
I've never done it, but a buoyed retrieval line as mentioned above is the other option. To me they seem more trouble than they are worth, but I have never lost an anchor (knock on wood).