There are two main bankruptcy chapters, so the answer to your question is, it depends.
A chapter 7 bankrutpcy is a snapshot of your finances at the time that you file. Debts and other legal liabilities that you incur after the BK filing are not considered.
You can only file and qualify for a chapter 7 BK once every eight years. So, if you file now and get a BK discharge, you will not be able to immediately file another chapter 7 just because you lost your appeal and have a new bunch of attorney fees or another judgment to pay. You might be able to do a chapter 13 filing if it has been at least four years since your chapter 7. As a practical matter, if you get this judgment discharged in BK, I personally would not bother with the costs and uncertainties of an appeal.
A chapter 13 filing is a repayment plan. You pay back some or all of your debts in a chapter 13 over a three to five year period. Generally, you need the permission of the court to incur new debt or make a significant outlay of cash for some purpose other than repayment of creditors. If you plan to appeal this judgment and you need to give an attorney a 10k retainer, you will need the permission of the BK court to do so. Whatever debts that you do not pay back in full are forgiven. For example, your chapter 13 plan might have you only paying back $20,000 of that $40,000 judgment against you.
Needless to say, getting the permission of the BK court for a 10k retainer, might be problematic. The US attorney's office/trustee will argue that the 10k should go to your creditors, not chasing after the "fool's gold of a successful appeal".
Before you say, I do not want a chapter 13, you need to understand that you must pass a means test for a chapter 7 filing and not everyone can do so. Moreover, if you do have a previous chapter 7 discharge that is less than 8 years old, then your only choice, in the event of further financial difficulties would be a chapter 13 filing or attempt to wait it out. 8 years, is a long time.