Your External IP Address (or, as you call it, the IP address reported by whatismyip.com) is not under your control and you can't simply set it to static. Your ISP gives you a new one when you reconnect, although there are chances that you might get the same, you should never count on it.
In order to solve this problem you have two choices:
Some Internet Service Providers (ISPs) offer you a "Static" IP service for a monthly fee. With the lack of IPv4 in the world, this service is unlikely to exist forever to individuals and it's usually reserved for companies and services that really need a static IP. Not all the ISPs offer this, you can always ask.
Use a Dynamic DNS provider. This will mean that although your IP address will eventually change when you disconnect (or your provider wants), you will always be able to get "back" to your Airport's IP by using a "name" rather than an IP. Any computer in your network will have a small piece of soft (some routers can do this automatically, I'm not sure about Airport Extreme, don't think so), that will periodically check your IP and if it has changed, it will inform the Dynamic DNS service that you use so it can update the value.
Some Dynamic DNS Providers (Mostly a free service) are (in no particular order):
Think of this as if instead of having a fixed phone number (static ip), a person could call you by "name" and the phone will check what is your current number (wouldn't that be great? ;) Nobody knows your number, only your name, because your number changes often (your ip!).
You haven't mentioned why you need a Static IP in the first place, but if you can't get it via your ISP, your only hope is to use Dynamic DNS. For what is worth, DDNS works really fine and it's very "fire and forget". Once it works, you no longer have to worry about IPs or anything, you just connect to: XXXXX.no-ip.org (for example)
The airport settings go by location, not which of the possible SSID you join, so make a home (or Jensen) location and use manual IP or DHCP with manual address to get things going.
Check your firewall as it can block DHCP responses, but I'd guess the router has issues before the Mac, but something is broken for it not to just work.
Best Answer
Unfortunately this depends entirely on the configuration of the router. Perhaps the DHCP (Dynamic Host Control Protocol) server is configured with a very short lease time for clients, and a new IP address is assigned whenever a client has no network activity for only a short time.
To fix the problem you will need to ask the administrator of the router to lengthen the lease time, or consider creating a reserved IP address for your MacBook Pro. The means of doing that vary with the make and model of the router.
After you have worked with the administrator to reserve an IP address, you will want to create a custom network profile on your MacBook Pro for use when you are connecting to that particular WiFi router, to disable DHCP (Dynamic Host Control Protocol), and configure a static IP address.
If all of the above is not possible, perhaps someone could propose a means of writing a script on the Mac that would cause the Mac, every few minutes, to renew its IP address lease to the DHCP server that the WiFi router uses, in order that the IP address assigned by DHCP would not expire, and the DHCP server would not automatically assign a new IP address.
TRY THIS TIP: Leave Apple Mail open all the time and configure it to automatically check for new mail every 60 seconds. This might be enough to keep your IP address renewed on the DHCP server of the WiFi router.