Hi Dimitra
Whether you can get costs from the insurance company or not depends on a number of things.
Firstly, do you have a contract or set of terms and conditions with them (This would only be the case if you were also a customer of theirs)?
If you do, then there may be details in the contract/terms and conditions detailing how far the company's liability goes in circumstances of fraud.
If not, then your only real avenue to pursue costs would be to make a civil claim against the insurer for damages. You could try this, in the first instance, by simply putting in writing what you request as full reimbursement for damages suffered. Then, if you do not get the outcome you like, you could lodge a claim in the small claims tribunal in your state.
For you to be able to get a damages award in a tribunal/court, you would need to be able to show that the insurer was negligent in their collection/use of the bank details and that there was causation between this negligence and the damages you suffered. This could be a tricky thing to prove, unless you can show there is a flaw in their process for collection and cross-checking account details etc or that there was another reason for which they should have been aware that the account details were being provided fraudulently.
So, in short, I'd recommend at least asking for the extra reimbursement for damages but keep in mind that there's not necessarily a definitive right for you to have them paid.
Best of luck and keep us posted with how it turns out!
Anna