I would leave the original transaction to go through. It may well be pending awaiting ticketing, though that's a circular argument. I would simply claim the direct cost of the Aer Lingus ticket on the basis that BA failed to ticket you, or contact you, ahead of check-in, and provided no way to contact them at the time of travel. You avoided IDB by your actions, so BA should be grateful that they were spared that expense.

ba.com/complaints
and consider that you may be waiting 6 weeks plus to hear more. Make the complaint first, wait for the email confirmation and then use the link to add in copies of your EI e-ticket and the proof of payment. I would consider this for MCOL if you want a speedier resolution.