You can know a person by their appearance, but not their heart.
Link wouldn't trust Elise Clark just because her attitude changed abruptly.
As mentioned before, Link took into account his relationship with the Serene Wizard and the fact that he didn't want someone with unpredictable behaviors and well-equipped to keep a close eye on him. That's why he chose to form a team.
What more could she do when under his watchful eyes?
At most, she could reveal his whereabouts.
But even if they didn't team up, with the Bionic Eagle, Elise Clark could still find him.
If she wanted to leak Link's whereabouts and set up an ambush, it made no difference whether they teamed up or not.
Moreover, if Elise Clark harbored ill-intent towards Link, she wouldn't have made such a farcical entrance.
She could have carefully designed a better way to gain Link's trust.
Now that they were a team, Link, of course, wouldn't show too much caution or alienation on the surface.
"Shall we divide the tasks?"