As K in Boston likes to say , income verification and fr are two complete beast. If you dont feel comfortable providing evidence of your income , always your peragative but like you mentioned, dont try to continue to get more credit from Amex. In my humble opinion, if someone want 50k in credit , just an example, from a creditor, they should be able to verify your income supports such a limit without a fight
You will not have any negative affect (related to relationship) with AMEX due to cancelling the account. A lot of AMEX customers cancel accounts and reopen them on a regular basis for a number of reasons. If AAOA is not a factor for you then I would call and cancel it. Now, since you are prepared to cancel the account, this is when you should try to get something from them. See if MR's or a CLI can be done as a retention effort. Let them know the reason you are closing the account and hint if it had been increased you would have probably kept the card. I know with the Delta branded cards, folks call to downgrade or cancel their cards and AMEX / Delta have been known to give miles, CLI and other things to keep the account. They could possibly say no, but it doesn't hurt to ask. Businesses love / need to have tenured accounts (especially when their are shareholders involved) so the really good companies always try to retain if they have had an account in good standings for a long duration.
Once again, all you can do is try. I wish we could guarrantee things, but hopefully this helps 