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I know this may sound too-simple, as a solution: but try setting the NAT feature on your Airport to Off, period.
That should take care of the 'double NAT' problem.
What are you trying to do with the Airport, exactly? set it up as a router, with WiFi, or something?
Try DHCP Only. Your ISP may not know enough about your local configuration or your aims, etc.
(A question: does the modem provide WiFi? if so, that would explain why the ISP warns you against DHCP Only mode.)
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That sounds extraordinary. I never heard of such a thing. Perhaps that is when the Airport is used as the modem as well as the router!
If I'm wrong, well, you may have to talk to your ISP about getting it corrected... or it may be as simple as 'set it to DHCP only, find out that internet is now Down, then set it to DHCP And NAT again, and get it working again'.
Are you willing to take the risk? I don't think it's much of a risk. I never heard of enabling DHCP requiring any such thing. Not with a router off a modem box, where the modem box is the only appliance around that actually 'sees' the ISP-assigned WAN IP Address.
BTW, what error condition do you get when you simply set to Bridge Mode ? I suspect some minor setting needs correcting for Bridge Mode to work.
What the other message is, may be quite illustrative of the problem.
You don't have any problem with DNS, or bringing up web pages, in any other mode with this Airport, do you?
Try DHCP mode. That may also straighten out the 'different IP address when connecting directly as opposed to when the Airport connects directly to modem with a cable' issue.
188.8.131.52 on the Airport, is OK. It's in Comcast's range of addresses (for Nashville region.)
184.108.40.206 is also Nashville region's Comcast address range. So you're OK both ways. It's just differently in Comcast's range, not a penalty or a problem.