They lack neither.Thats the thing, they lack that. She is part of the guild, a group not bound to or by thier nation. It would be like capturing a noble from another nation, and telling that national to respect your judgment as you blame your own fault on that noble.
Sure your nation might buy it, but the guild won't. And people will still ask why is this one person, who wasn't in charge or anything responsible while the rest are not. There is already a faction divide. This just pushes the guild to side with the other side.
Its also pointless, just play off being guards that failed, I mean really how would you even avoid or defend vs what happened? They can just act like they were ambushed. Sure its a blemish on their records till their faction takes power, but it wont affect them as its just dismissed.
This is a medieval type setting, and with that in mind:
I think you underestimate just how much power religious organisations can wield.
I also think you're overestimating the standing of someone who's basically middle management of what is likely a commoners guild. Certainly nowhere near that of even a minor noble.
If Amelia gets blamed, charged and punished for the loss of the Saintess (someone we can assume is hugely important, even if WE know she's a fake), I could see the Royal Capital, at best, issuing a sternly worded letter to the church.
Now I don't recall if the manga went over the power or structure of the adventurers guild in this setting (is it multi-national?, is it state or noble backed? etc.), so their level of influence and power is unknown. Going by both other manga in this genre AND real world historical medieval guilds, they might only exist at the whim of the authorities, or they could actually wield considerable power. Even if it's the latter, the Guild might think that starting a fight with what's presumably a larger, more influential and powerful organisation might not be worth it. Especially over the life of a secretary and a random new adventurer who are being blamed for the death of the Saintess.