I'm wondering if airline will claim you took a voluntary denied boarding or were denied due to documentation.

What you should have done is demand rebooking on the other airline on spot, or a denial in writing.

In any case, see what your TA will say, and then claim either IDB compensation or compensation plus full refund from the airline directly.

You *can* claim the other ticket but it's unlikely the airline will pay.

Do also file a complaint with the relevant National Enforcement Body to the fact you were not properly informed of your rights while being IDBed.