Given the very long timelines in making an animated show, Prodigy’s storyline for the season would have been done, and in this case the dialogue was recorded, while Picard seasons three was still being developed.
We know that Kurtzman instituted monthly meetings to short out plots and use of legacy characters sometime while Prodigy was producing the first season because there were mentions in interviews about sorting out the use of Okona between Lower Decks and Prodigy.
Asking why Picard was able to go forward with a very similar storyline is reasonable. One inference is that somehow those in charge hadn’t really taken into account the overlap in audience between a live action adult targeted show like Picard and Prodigy as a family oriented offering.