You know, that’s not a bad idea. Issue with Jar Jar is that nobody really meaningfully interacts with him. And he’s still an idiot. But what if he’s on a kind of personal quest, kind of like the Quarian pilgrimage from Mass Effect, where they leave the tribe in order to find and bring back something of value? And what if he- perhaps in an extremely naive way- decides he’s going to seek Padmé so he can make peace between the Naboo and the Gungans?
It’s a very naive, overly optimistic thing, which the audience and other characters can pass off as being silly, and they can completely dismiss him.
But, it would explain why he’s eager to join/guide the Jedi to Padmé, why he sticks around, and ultimately why he suggests the Gungan army to Padmé.
In his scene where he meets Obi-Wan and Qui-Gon, he could clumsily explain part of this - “Mesa on a personal quest, but then thesa droids attacking, and mesa getting lost!”, into leading them to Theed.
In the scene where Padmé asks Jar Jar why he’s still with them, he could explain it more clearly- that he’s seeking her because he wants to make peace. Her facial expressions there give the impression that she thinks he’s cute- but dismissable.
Padmé and Jar Jar have a more serious conversation on Coruscant about their peoples, which also could include a little extra.
Then, you wouldn’t need to do much more- Padmé and the Gungan king would talk as before, and at the end he’d declare “Peace!” to Jar Jar’s cheers.
