If you "fail" a FR, AmEx will close all of your cards; closing the one card that triggered it will not help.

If you know that you will fail a FR, you may as well close the cards yourself, that way at least it will not show on your Credit Report as "Closed by institution".

Once AmEx closes your accounts, you have 90 days to use your MR points, and it will only be doable on phone, online you won't be able to access your points. (PM me with email if you want me to send you a scan from AmEx saying as much).

If you just want to buy time on an FR, while you are trying to get the points out, fill out the 4506T form as best you can, but put a different address than your own. It will buy you about two weeks.

As of October 1st Federal Law requires that CC companies do not count "household" income, so even with your explanation, with a $5k income you are almost certain to fail the FR. (See here for report on change.)

Don't ask how I know all of this!

GL with your FR!