Using a capacitor across the welding supply will not help much, the cap will discharge in next to no time and you will still have some ripple on the welding current. The capacitor current will be huge and probably shorten the life of the cap(s) considerably. A ball park value could be 100,000uF or larger with a 1000A ripple rating (if such a beast is manufactured).
An inductor in series is essential to smooth out the welding current but with a single phase rectified power source you have little to no chance of getting a continuous output current from your welding set.
Big welding sets were once built using a motor generator set that gave good smoothish DC for the welding supply, it is immaterial what the prime source of power (single phase motor, 3 phase or IC engine) is in these sets since the output current comes from the torque applied to the generator shaft.
A three phase supply can be rectified and produce an EMF that only falls to 86% of the peak AC voltage at 6 times the input frequency (50/60Hz in ,300/360Hz out)this in association with an inductor (choke) will provide a suitable source of welding current for DC welding.
If the same rectifier/filter is used with a single phase input the source EMF will fall to zero twice per input cycle and result in a discontiuous weld current.
By installing an inductor you will improve the smoothness but will not eliminate the high ripple current, an increse in rectifier input voltage will be required to accomodate the voltage drop over the inductor.
If you can use a 3 phase input for the power source if you require a very smooth welding current. Its a wee bit more complex but it will work.