I'm trying to visualize that. Are you somehow accidentally short-circuit wiring something there?
Are you sure that what needs to be wired in parallel or in series, is wired that way?
It sure sounds like your LED doorbell might be causing the problem. I'm just wondering if you can
eliminate your wiring scheme as a potential cause.
The relay coil needs to be energized with 24VAC when the doorbell push-button is depressed (a series circuit), and the
contact closures on the relay should then trip your camera trigger, (however that circut is wired).
I know you're getting relay contact chatter. That can easily be solved with a diode or MOV (metal oxide varistor) across the coil.
My understanding of your post may not be correct. Can you just touch the two wires across your doorbell contacts to complete the circut?
(temporarily eliminating the LED in doorbell from the circuit altogether for this test), and see if you're still getting relay chatter.
Something just doesn't sound right to me about this, but I may not fully understand the problem.
Good luck.