Do you have that chat history in writing? It would be important. If not, ask them once more and take screenshots so you can prove their denial, which is against EC261.

You may proceed to book new flights and then, after the flights, send the bill to AY. Forget about lounges and exit seats, it is the flights that count. (If I were you, I would pick #2.)

It is unlikely that they will pay without a fight, though, so be prepared. Moreover, it is my understanding that you shall not cancel the AY reservation as in accepting the cancellation you waive your rerouting rights.