A319 Captain reported during climb after takeoff an ECAM ANTI ICE CAPT TAT was received. The flight crew returned to departure airport.

Narrative

Light rain in ZZZ. Takeoff climbing out of about 2;000 feet. got an ECAM ANTI ICE CAPT TAT. Leveled off at 5;000 feet to work ECAM. Called Dispatch on iPad phone app; but did not work. Tried dispatch VHF; no answer. Then call Maintenance at ZZZ to call Maintenance Control and Dispatch to relay a message; took long time - did not help. Then used radio and got phone patch to Dispatch and Maintenance Control . After all of this I burned fuel; returned to ZZZ to refuel and get an MEL. ATC ask if we want to [request priority handling]. I said ok.The problem was my bias. I was climbing into icing conditions and ZZZ1 had low clouds with potential icing. My thought was I don't want to fly an aircraft into icing with into TAT probe inop. The ECAM had no actions and on status page was no notes. In QRH ETOPS said continue with concurrence. Because there were no ECAM actions; I should have looked at it as crew awareness no immediate threat. No message on status page to avoid icing. I should have continued the climb into icing and later send a msg. to Maintenance Control/Dispatch. The phone app did not work at 5;000 feet that slowed down my communication with dispatch. I will study the ECAM procedures. I am used to doing ECAM actions and not any action through me off.