Jim,
The very minimal amount of sulfur (0.034%) over the allowable amount in a 304 or 304L (0.030% max.) should have virtually no noticable affect on the weldability. In my opinion, it is undetectable. We must realize that the effects of sulfur are to improve machinability, but it also promotes hot cracking during welding. In GTAW, sulfur does increase weld penetration, but that, too, is not a weldability issue. A 304 or a 304L, with allowable amounts of sulfur, very seldom have any detrimental affects on the welding of this grade of stainless steel. The only "bad" thing of sulfur in a 304 grade, is the slight possibility of hot cracking, but when was the last time anyone has experienced that problem with a 304 using acceptable welding procedures? The amount you stated (0.034%) is only 0.004% higher than the allowable amount of 0.030% for a 304 or 304L. This is almost a trace amount, and it is my opinion that it would not, in virtually any circumstance, have a noticable affect on weldability. On the same hand, we should never say never. Improper welding parameters can possibly punctuate the effects sulfur might present itself.
Chuck