I replied your post previously but without the following details.
My situation is similar to yours but there may be a BIG difference: initially my lawyer did not guarantee that he can had my case done before Oct although he said he would try his best. I had to do most of the work to have my case done before 10/1, he submitted the application for me in the end. I was very upset for the service and argued that I should pay less, but he argued that that was because of my push and he could handle all the cases like mine in a short time. Because my contract was for the cost of the package, not the hours worked, I paid the fees in the end to get rid of any potential trouble.
Please keep us posted if you can get a better deal in your situation!