So I wasn't wrong, perhaps I could have chosen a different word other than bank, but you should always be able to make a complaint about a payment from both directions, you had poked the target (Qatar) and they were having none of it, so it's now time to go after the source, Ideal.

Also, it goes to highlight that putting a 3rd party in any contract can make it more cumbersome if something goes wrong. frequent flyers will often say you should usually book direct with the airline and avoid booking tickets with a 3rd party. The same can also apply with payments, adding 3rd party payment processors such as PayPal or in this case Ideal can make it a bit more tricky if something goes wrong, albeit payments will likely have more consumer protection

Hopefully you get reunited with your money before too long.