If the Jedi begin on their mission to broker peace between Gungans and Naboo, then Jar Jar could simply be their go-between/guide/facilitator. (He could even be their Representative, which has supporting voice lines.) You also don’t necessarily need to have them leave in the bongo - after the Otoh Gunga scenes you could potentially cut to the second half of the scene where they originally meet Jar Jar, where the Jedi and Jar Jar talk more (ending with, for example, my ‘we must hurry’, eventually cutting back to them appearing in Theed having travelled there offscreen.)
I think I’ve got an idea for a good opening that does exactly that. Something like this:
-Crawl
-Jedi traveling to Otoh-Gunga
-Politicians in Theed, comunication disruption
-Trade Federation arrives above Naboo and start an invasion immediately
-Qui-Gon and company are surprised by battle droids while on their way to theed
I think I’ll have a rough cut ready next week.