In 1 Kings 2:13-25 why didn't Solomon's mother see Adonijah's request as treasonous?
Because she is simple and naive, even dumb and stupid.
She bathed where David could see her in 2 Samuel 11:2
One evening David got up from his bed and walked around on the roof of the palace. From the roof he saw a woman bathing. The woman was very beautiful,
She married one of the most righteous men in the Bible, Uriah the Hittite. Yet she was unfaithful to him in 2 Samuel 11:4
Then David sent messengers to get her. She came to him, and he slept with her. (Now she was purifying herself from her monthly uncleanness.) Then she went back home.
as if this was no big deal. There was no mention of her reluctance or regret.
After David eliminated Uriah, she married the man who gave order to kill her loyal husband. The Bible didn't say whether she was aware of David's evil plot. In any case, I don't think she cared because she was not thinking or analyzing. She was a simpleton.
Adonijah was a smart guy and knew that she was naive and approached her in 1 Kings 2
13Now Adonijah, the son of Haggith, went to Bathsheba, Solomon’s mother. Bathsheba asked him, “Do you come peacefully?”
He answered, “Yes, peacefully.” 14Then he added, “I have something to say to you.”
“You may say it,” she replied.
15“As you know,” he said, “the kingdom was mine. All Israel looked to me as their king.
She didn't react to Adonijah's claim "as you know" and stupidly granted Adonijah's request to ask Solomon on Adonijah's behalf.
There was no record of her making a decision on her own and she seemed to have trouble of saying no to anyone.
At best she was naive, at worst she was stupid. I think she was both.