Here’s an example of how Jar Jar could care about and be more actively involved in the Queen’s rescue. This’d be more for the scenario where we skip Otoh Gungah because Jar Jar is their guide - albeit a bumbling junior politician who’s uncomfortable in dangerous situations. And Qui-Gon is initially hesitant to trust someone who’s clearly a bit out of their depth, ultimately accepting his value given his local/political knowledge. At the end of this clip, Jar Jar tries to big up his contribution as he realises he wants to be a part of this group and that the Jedi are competent protectors.

Again, it’s rough, but should illustrate some of our options!