Ramban comments that Hashem told Moshe Rabeinu to say the particular phrase pokad pokaditi to the elders, and that there was a tradition passed down from Yoseif to his brothers that the redeemer would use this phrase, but that Moshe could not have known of this phrase because he was taken from his father's house at age 12.

Along with the tradition apparently it was known that it was to be kept secret that only the elders should know.

So while Amram certainly could have known this, Moshe, not being among the elders when he left Egypt, would not have learned it from his father.

However if that were truly significant to prove that Moshe could not have known, I have a problem, since Aharon HaKohen by the time he was sent to meet Moshe was likely among the elders and certainly old enough to have learned it.

A skeptical elder who would have otherwise been convinced of Moshe's legitimacy due to Moshe not having had a chance to have learned this secret from / among the elders, particularly if/when Aharon is Moshe's spokesperson to them, cannot be certain that Aharon didn't expose Moshe to the secret value of this phrase!

Thus, if we assign a human trust level of confidence that the elders would have in everything Aharon would say as a kosher Yid, we still cannot assign a miraculous secret message priority level of confidence to Moshe's saying that particular message proving that he was the predicted redeemer!

Please recall the 3 signs which Moshe was given to perform for the elders... saying Hashem's name, throwing down his staff to turn into a snake, and putting his fist to his chest...

It seems that the Torah says that if they don't believe the first sign, then they will believe the second sign; and if they still don't believe with the second sign, they will believe with the third sign.

Thus we should be sensitive to how a skeptical audience would view signs of Moshe's authenticity.

Within this skeptical attitude, perhaps Aharon, who as a prophet "usually sent" on missions such as Moshe did not want to accept, and sent to meet Moshe (and remembering Miriam's prophecy, and Moshe's special "goodness" at birth through the reed basket incident) -- Aharon surely needed no convincing. Whether Hashem told Aharon that Moshe was the redeemer, or whether Aharon recognized Moshe's believability when they met is immaterial. Moshe saying "pakod pakoditi" to Aharon in the absence of the elders would only have verified to Aharon what Aharon already knew at a miraculous level of confidence.

But even with Aharon as a recognized leader and prophet among the people, is there any limit to his believability to the people?

If there is no limit, then no signs would have been necessary at all - Moshe only needed to convince Aharon that Miriam's prophecy was accurate, or that he would know prophetically, or any other way for Aharon to be convinced -- and if Aharon told the people that Moshe is the redeemer, that's sufficient.

But if there is a legitimate limit, particularly when the Jewish people are going to have to endure their lives as slaves getting harder without bricks, and an earlier misguided attempt to leave Egypt early, then how does this particular sign -- Moshe saying "pakod pakoditi" after Aharon had himself become convinced -- how does it carry sufficient believability weight any more than Aharon's own being convinced?

...

Update - this coded phrase was surely convincing to Aharon himself, not that there is any indication that he needed any.

But Aharon's credibility with the elders and the Jewish people was such that if Aharon said that Moshe said this coded phrase, that was sufficient for them to believe that Moshe said this coded phrase on his own.

And briefly, perhaps if any of the Jewish people had the slightest doubt as to whether Aharon had made this up, perhaps Korach would have raised this question. If even Korach didn't ask, and since he did make it out of Egypt perhaps we know he didn't ask, then perhaps noone had any question about such a thing.