When a client makes a via GTM, it starts with the DNS request to resolve a hostname to an IP address. There hasn't been an HTTP request at that point, so you wouldn't be able to consider HTTP cookies yet.
Once the client makes an HTTP request, you could track which datacenter they'd been resolved to using a cookie. If they flip datacenters mid-session, you could redirect them to a hostname that points just to one datacenter. You could also consider using an iSession tunnel to route them directly to the other datacenter.
Aaron
Nimbostratus
