Is GTM really needed for your case?
I mean if you setup lets say 10.0.0.0/24 to be used for VSERVER ips at each site you could then for site1 setup 10.0.0.1 as VSERVER which will have the server1 from site1 as first choice and then server1 from site2 along with server1 from site3 in the same pool. Setup the same at site2 and site3 (but favour the local server to be used as first option).
This way the client traffic will pass the F5 at the same site as the client where the F5 will then take a decision on which server will be choiced to handle the request.
Like so:
Client at site1 -> F5 at site1 (LTM decision) -> server1 at site1
or if server1 at site1 is failing:
Client at site1 -> F5 at site1 (LTM decision) -> server1 at site3
Where client at site3 will have the following logical flow:
Client at site3 -> F5 at site3 (LTM decision) -> server1 at site3