Would accept their “gag order” if they had agreed to pay you what you asked in the first place? If not, why didn’t you ask for what you really want. I understand that you feel they screwed you over, but they really can’t change that, now. All they can do now is limit the damage.
In all likelihood, you’re no longer dealing with the incompetence that caused the problem, but with a group who has a lot more experience with this than you. Obviously, they will pay more than what they offered (otherwise, they would have offered too much, and if they’re experienced, they aren’t likely to do that). You need to decide what settlement you would accept and politely and sincerely tell them that is what you want.
If what you want now is different from what you initially asked, you will need some justification for changing your mind. If your justification is because you want not only for them to reimburse you for the extra expense they caused you, but you want to continue to make claims against them and complain how they did not satisfy you (even though they did everything you asked), I don’t think you will go very far.
It seems your best bet is to inform them that you expect them to reimburse you for the costs you incurred because of their incompetence and if they do, you would consider the matter closed. No additional claims, no additional complaints. Of course, before you do this, check your belongings and make sure they are safe and not damaged/missing as once they settle with you, they aren’t going to do anything else.
I’d guess you’ve got a 50:50 chance of them reimbursing you for the full amount. If so, I’d feel that you’d won. You aren’t likely to profit from their mistake. If they cover what their mistake cost you, what do you have to complain about?