Euphie's offer was basically reaching Anis darkest desires, that was to live for herself selfishly without a care. Anis magicollogy was originally created to solve her magic disability, that it could also benefit others was a side effect not the main objective of her studies, shes not a bad person but not a saint either.
She wanted validation and to live for herself not as a ruler and her actions indirectly caused Al's big clusterfuck and showed her in the face the consequences of her selfish actions.
As stated before Anis is not a bad person just a selfish one, but even her was forced to realize that with Al gone, she must take on her original duties due to be the only remaining child and also as a self atonement for her selfishness and carelesness
Imagine Anis might have felt when Euphie offered not to take her own burden but also to stop being human just for her sake, she was not only angered at how self sacrifical Euphie was willing to go, even more than Al himself, but also intense self loathing because probably a small dark part of herself did actually desire for it, not mentioning that Anis is aware that she a faliure magic wise, that she will be a faliure ruler wise, but when even her bloodbirth's right is taken away, she would lose not only her wishes but also her regrets as well, she would have "nothing left"