First, try seeing if the idle current really can blow a 3A fuse. Or whether it can heat up short lengths of different gauge wire. This should prove whether genuine power is really being wasted.
Also, is the transformer in decent shape? Too much current, or too long current, can overheat it. Varnish insulation might break down. Metal plates might separate, creating a bulging transformer. Windings might make electrical contact with one another. Greater idle current (wasted power). The transformer suffers a permanent decline in performance.
A visual inspection may reveal transformer damage. Check coil isolation, winding isolation, etc.
Can you please let me know whether is there a way to switch on the transformer when the welder touches the work piece with the electrode?
My suggestion:
A welder wants to get immediate arc simply by touching the rod to the work. (I have experience with a 230 amp AC welder.)
Your circuit shall be like a watchdog, sensing immediately when he touches rod to work.
This can be done by sensing a change in voltage or current, in the secondary coil. (Theoretically voltage should fall across the secondary when the rod touches work and completes the circuit.)
Your sense circuit will then close a relay on the mains. (You might get by with a beefy triac.)
Primary coil gets power. Secondary coil gets power. Welder gets immediate arc (or almost immediate).
While he welds, your sense circuit holds the relay closed all that time.
When he stops welding, the secondary goes quiescent again.
Your sense circuit opens the relay after a while. It returns to watchdog mode.
So now everyone says "But if the relay is open then there's no voltage on the secondary for the sense circuit to sense."
My answer is that it should be possible to supply auxiliary power to the secondary.
Perhaps by attaching a second transformer to the main one inside the welder. It may even come to winding a hundred turns on the main core to create an auxiliary primary.
This is just brainstorming.
There will be a lot of testing required to see what electrical havoc takes place in the secondary during welding. Testing to see how robust the sense circuit needs to be. Etc.