Will a tvs diode help my relay with inductive loads?
I think their concern is that with the inductive load, when you open the relay it will force some current across the open contacts and spark, damaging the contacts. A TVS diode should protect you, and you'd need to choose values above the mains voltage (peak voltage, not rms). In this case you'd want >170V. This wouldn't protect it 100%. I think you'd still have some current forced across the gap a tiny fraction of a second after opening it (before the voltage across the relay surpasses 170). So I can't say how much above 3A you'd be gaining.

The other option would be a bigger relay of course.