Here is a couple of things to consider. These may seem kind of contradictory but its just food for thought.

1.) If it doesn't make it into your PRIA records, you could consider just not listing PSA as a former employer on an application. You were only there a few weeks.

2.) Even if it doesn't make it on to your PRIA records, I suppose its all in how your next employer asks the question. If they simply ask "have you ever failed a checkride" then you can answer no. But if they ask it like "have you ever failed a checkride, dropped out of training, or been terminated or asked to leave a flying job" then you might have to fess up, assuming of course there is something on your record to report.

I don't know what the requirements are for airline employers but my guess is the FAA will want them to report quickly so the information is available to the next employer. But, in the end, if there is nothing on your record to report, I would not report it.

Setting all that aside for a moment, maybe milkman above is correct and you should consider a carrier that flies turboprops.