Exactly, Awakening has a significant jump in power, so Alicia would have to awaken to fight the Abyssal Ones regardless.
As for Teresa, she may have hid her hand from the Organization, but judging by the 3rd data book, they were well aware of many things that we did not believe they knew about, they even know about Clare's 4-limbed Partial-awakening judging by the data book.
I also doubt that they do not check whether a warrior has been confirmed eliminated or not. So Teresa's rather unbelievable statement over Rosemary NOT awakening was probably seen through immediatly.
Especially considering that Teresa claimed normal yoma had caused her injuries...pfff yeah right.
I also had a thought on how Alicia's stats can be so high, technically she is a partially awakened #1, and as we have seen, the partial awakenings increase power significantly.