Well as an external there are 2 things I don't understand.
1/ This case is fishy as hell. Dude can't you call your "brother" and set up things together? Seems like an obvious thing to do for me. And if you shared the account then the fault is on you anyway.
2/ Why in hell should Bitdouble refund you?
Are they at fault in any way here?
Did you use all security tools they put at your disposal?
Answers of all above questions are clearly no.
BTC is full of scams and hacked and you should be aware of it. You didn't secure your account completely so you're at fault.
Even if what you say is true even if you do own the adress and whatever you like, I still fail to see how Bitdouble should pay for your mistake: not securing your account.
They should do it if you hadn't the possibility to use high security measures, but here you had the possibility. You were just too lazy to use those tools.
I maintain my points. I read the case and I don't see what could they be accused of.
Here, refunding you is a pure commercial gesture.
They CAN do it if they like it, but you have nothing to claim. Of course I'm sure you would have appreciated a refund and it might have given them a generous reputation. But you are the one to blame concerning the security of your account here.