I had a thought of using it on our tank's shield as it is explicitly indestructible,
Not sure what rule you are using to determine that.
The rules effect of an infernal contract is that of a magic item with the contract trait. This item has no physical substance and can't be dispelled, altered, or destroyed except by destroying both physical copies of the contract (a virtually impossible task, as most devils send their copy to the Fallen Fastness in Dis, an infernal library all but off-limits to mortals).
I'm not entirely sure that you could pickpocket the mortal copy of an Infernal Contract and have it be usable as the magical item that it is.
But the contract item is itself very much destructible. You can't break the contract effect by destroying the item. You have to destroy both copies.
The Contract Devil was defeated and sent back to the Hells
I'm afraid I can't help directly with the main question, which seems firmly in the "What you think would be most interesting/fun" camp, but I've noted what may be a minor lore misconception.
Unless the party defeated the contract demon by banishing them back to Hell, they are probably dead. On Golarion and the Great Beyond in general, creatures like demons and devils do not merely return to their home plane when slain. It's not really clear to me what happens to the body, but the essence of the creature returns to its home in much the same way the matter of a deer returns to the earth: In a decidedly inert state.
For the majority of immortals slain off-plane, this usually means getting as far as the Maelstrom before being sucked into its great soulstuff recycling vortex. If that contract devil was slain, they're gone unless resurrected. Even if the soulstuff got back to Hell in one (more likely several) piece, it's going to go toward fertilizing the Hellish ecosystem.
Personally I would argue that such a contract is destroyed, or else rendered destroyable on death of the contract holder... though on the other hand that might interfere with the fantasy of such contracts - at least signed ones.
EDIT: Hold up, obvious key words, "A signed contract carried by a living contract devil." - Even if they merely got banished, they're not carrying the contract anymore.
It very much seems to be a physical object and it is worn on the Devil. And as it says, it is immune to all damage, hence indestructible except from that Devil.
I'm not entirely convinced that the item is indestructible, but that is a bit beside the point. This isn't the rules forum. So rather than keep arguing about it, I'll just roll with that interpretation.
I think you are right that making the shield indestructible by putting the scroll over it may be a bit too much. But not by a lot. There is already an Indestructible Shield. Using that as a reference for the combination of regular shield with contract attached would probably work fine. It wouldn't completely block all damage from reaching the character - it would still be using the shield's hardness as damage reduction. You just wouldn't have to keep repairing the shield after every fight.
You could use it as a prop for Earn Income. Take it to a tavern and use Deception/Diplomacy/Lore skill to con people into taking a bet that they can't destroy it.
On Golarion and the Great Beyond in general, creatures like demons and devils do not merely return to their home plane when slain. It's not really clear to me what happens to the body, but the essence of the creature returns to its home in much the same way the matter of a deer returns to the earth: In a decidedly inert state.As I understand, a body of 'spiritual' creatures is spirit essence in 'material' form ('quintessence' I believe it was called). And if the creature is dead, its body can't exist normally in material world and just disappears.
Oh certainly, and that would have been my first assertion, too, but I seem to recall enough references to demon corpses and items made from them to cast doubt on the idea that these quintessence bodies necessarily vanish in short order. I could be misremembering, and I don't have time to look for them now, but it seems like these immortal spirits decompose a lot more like material corpses than my first expectation (even if the body is fully made of quintessence, ofc, so this decomp involves the spirit essence likely returning to the Maelstrom in its order, too)
I think that the concept of 'extraplanar creatures that are killed on the material plane are just banished back to their home plane rather than actually killed' is a D&D-ism that Pathfinder doesn't necessarily hold to. And probably will do so less with the split from the OGL.
Secrets of Magic had a sidebar lore entry about how Summoning spell creatures are magical constructs rather than existing creatures that were teleported. I think there is also something about outsiders using Plane Shift or the like to actually travel to the material plane and so if they die they actually die rather than get banished. Sprite PCs for example should probably die when killed rather than get banished to the First World.
I am talking about this from the Phistophilus
Ward Contract: A signed contract carried by a living contract devil (including draped over its horns) is immune to damage from all creatures other than that contract devil. A contract devil is immune to mental effects that would make it destroy, nullify, or alter a contract.
It very much seems to be a physical object and it is worn on the Devil. And as it says, it is immune to all damage, hence indestructible except from that Devil.
That is an ability that allows the devil to protect the contracts in their possession.
If the contract is not in their possession, is not signed, is subject to damage that did not originate from a creature, is subject to a non-damaging effect that can destroy items, or if the contract devil is not alive, then the contracts are not considered indestructible.
So draping the contract over your friend's shield is unlikely to afford any more protection than regular parchment.
You really must read the fine print my good fellow.*