Buster Sword?

Planted for a fallen memory
I'm pretty sure it's planted in the ground at some point to commemorate Zach.
It's fair to say it's in the story
Well, definitely.
A shattered weapon lottery
Lottery because it seems to go from person to person, almost whoever wants it gets it. Also shattered perhaps due to how it splits up or something?
And super sticky inventory
Impossible to get rid of the Buster Sword from your inventory.
An image asks you to continue?
Is it on a Continue page somewhere? Dunno.
Final resting place a church
Probably at the end of Advent Children.
Or did it find another venue
In a business district perch
While its church is renovating
D:? Dunno.
Twelve Gilgamesh is hating
Gilgamesh has a Buster Sword in FFXII but it has a mark on it that means 'fake'.
From angel who is irritating
To the man precipitating.
D:? Dunno.