I'll make some comments on your posting.
SMAW (popularly called stick welding) is an approved method of welding, so no official code or standard (ASME, AWS, API etc.) will forbid to use it.
I seriously doubt that a pressure as low as 5 inches (125 mm) of water column could cause a Schedule 10 ss piping to collapse due to internal pressure. If it failed, this must have been due to structural reasons, such as a long, unsupported run, for example. In this case, the welding had nothing to do with the collapse.
Back in my days of erector engineer (many many years ago) I used to make hundreds of stainless steel weldings using stick electrodes, so personally I have nothing against them.
If the WPS issued by your company allows SMAW in ss provided a backing strip is used, then your client has the right of requiring that the welding is done with stick electrode and backing strip.
Having said so, I'll give you a couple of arguments you can use to convince your client to use GTAW, popularly known as TIG welding.
1st. Time delay. Clients don't like time delay, actually they hate it. You can say that, since your welders are not qualified for SMAW, they must be qualified, and this will take time.
You can go on saying that SMAW is slower than TIG, because you have to remove the slag after each pass using aluminum oxyde grinding disks (never use silicon carbide on ss !!!) and ss wire brushes (never use carbon steel wire brushes on ss !!!), which will cause unforeseen delays in the work because TIG doesn't produce slag.
IS YOUR CLIENT PREPARED TO GIVE YOU A JUSTIFIED TIME EXTENSION ??
2nd. Additional costs. Clients hate also additional costs. You can say that, since your welders will have to pass the test for stick electrode, they'll have to be trained in that method, and this will take a few days "burning electrodes" in an unproductive job.
You can go on saying that the delay caused by SMAW instead of TIG, as explained above, means more manhours than those originally calculated.
AND WHO'S GOING TO PAY FOR THAT ??? Your company? Nooooo, your client, because it's him who's requiring to follow a different method than that originally established.
Try those arguments and let me know whether they worked.
Good luck
Giovanni S. Crisi
Sao Paulo - Brazil