It may seem odd, but, your smaller wire will require MORE current to create the heat creates the weld. You're relying on the resistance of the wire to create heat
Heating power = I * I * R
The resistance of the smaller wire is about 64% of the resistance of the bigger wire (by cross-section) so you'll need about 25% more current to reach the same power. But, you'll need to apply the pulse for less time so you don't melt the wire into a puddle. You'll have to test it a bit on pulse length.