Ah, the ages old nurse in the middle and getting all the blame scenario.
I would have done the same as you did. I would have probably used the biopatch, although to be honest, if you maintained your sterile field the chances of infection from not using one, especially given that the dressing was likely to be removed in the near future and replaced if the line needed to be advanced, were probably slim.
As far as it not working anymore, was that before or after they tried to manipulate it back into place? If they just left it in the same place, I would suspect that that could be the reason why it wasn't working, given that the ED doc said it was not in proper placement. Either way, the doc who actually looked at the CXR for placement is the one whose opinion I would defer to. Unless the other doc viewed the film from home before giving the go ahead to use the line...but I would hope he would have told you that he had looked at it before making the decision.
In this situation, it might be in your favor to drop an email or other written account of what occurred to your NM, in an emotionless and factual way. Like instead of saying "dr. was angry" put something like "MD objected to the decision to not utilize PICC line despite prior orders from ED MD to not use line until CXR viewed by primary to maintain patient safety secondary to questions regarding placement of PICC." Be sure, if you do do this, to include all steps that you took regarding the decision and to maintain patient safety in this manner.
Ultimately, yes, the kiddo did get another line. However, they also got their abx into them, and in a way in which there was no question as far as delivery into the right place. We are in no position to judge the risk vs. benefit of what the ultimate outcome was, which is why you deferred to a MD and then used that information to make your final decision, and why the docs need to take it up with each other if there's a discrepancy in opinion.