DNS request can provide a single IP? and then change it to some other IP?

The easiest and only correct solution is for the customer to allow both IPs, as they produce the same service but at different times.

On another note, with DNS based load balancing, this is a common issue. Mainly because the FW policy builder will resolve the FQDN at the time of policy verification/installation, and after that only the one IP will be used.

You could probably set up a LTM VS for that customer in particular on one of the sites, with members from both datacenters and monitors to actively disable the passive site, but that would again leave the one BIG-IP as a single point of failure.