actually you can reenter Kilika cloister of trials much sooner, namely once you finnish the events at Djose temple (after acquiring Ixion)
It is the Fayth Chamber at the end of the trials with the extra items inside that you can only access after the highbridge scene (the one Yuna always goes into alone to pray for a new Aeon at each temple) this is the case for all the temples fayth chambers
on a side note about revisiting the trials for the destruction spheres :
you cannot backtrack early to Besaid trials like you do at Kilika since Isaaru the summoner is using it up until until after Dark Valefor appears after the highbridge scene, meaning you cant get Anima before killing Dark Valefor if you missed it during your first visit
you can reenter Djose temple as early as upon reaching the thunderplains
Macalania temple can be revisited after you leave bevelle and get dropped off in the woods by the airship (just before the calm lands) but Dark Shiva will already be blocking the way
you cannot ever reenter bevelle temple but you also cannot miss its destruction sphere, thank god cause that mini-game is lame, ill note the extra chest with the spear for Kimahri in the bevelle trials has nothing to do with the destruction sphere
also its the opening of the chest itself that triggers the Anima Locks, the item inside has nothing to do with it
just to reiterate it is only the fayth chambers that are opened by the highbridge scene, dark Aeons appear at different times and each temples cloisters are reopened at different times too
two years late reply but i hope this helps someone as ive found there is alot of confusion and wrong information out there