A lot of interesting thoughts here.

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.)

Another option to preserve Otoh Gunga, though more radical and mutually exclusive from the above ideas, would be to have the Jedi visit the city upon their RETURN to Coruscant, failing to convince Boss Nass to help them but at least convincing him to meet the Queen, transitioning then into the Queen-meets-Boss-Nass scenes where she wins him over.