You made a statement "We would like the GTM to always hand out the same IP, regardless of which data center is active". Wouldn't that be the definition of a static entry in DNS? If you add a static A record it would do that. I feel like you mean something else.
Solution 1 (not so great) The GTM can hand out two IP addresses at the same time, but there is no guarantee which one the client uses in which order. That is the problem with handing out two simultaneously.
Solution 2 (good) You can combine the GTM with BIG-IP Local Traffic Managers doing Route Health Injection to accomplish a scenario where the IP address moves (via BGP, RIP, OSPF) when all services behind that vip become unavailable. Route Health Injection means that if a service goes away, BIG-IP at site1 withdraws the route for that virtual and your routers in the network will push traffic to the secondary site still pointing at the same IP and the client does not need to re-request. This requires BIG-IP LTM at both sites with routing modules.