Are you comfortable revealing WHERE in Manhattan the summer intensive is - so one can gauge distances to be gapped?
My feeling is that you’re in NYC anyway - so you might as well attend the info session, as long as your daughter doesn’t feel you’re infringing on her process/experiences. You might learn about some angles of the college (favorable or unfavorable in your mind, that you can report to your daughter to see how she feels about those).
Then use “Find Your Phone” or similar tool so that your daughter might catch up with you along the way. Make sure she brings a copy of whatever paper she needs for campus access - although she will likely have to wait outside whatever building you are at the time so you/the guide can “fetch” her.
Since the “Barnumbia” campus is so completely unique both from NYU and from Fordham, I think she would definitely benefit from laying eyes on it in person, eventually crossing the street to the main university campus (as she would many times while student at Columbia University, for academics, certain libraries, certain dining halls, gym, hanging out with CC friends, certain clubs, etc.).
Later stroll down (south) on Broadway, possibly catching a bite, while (unknowingly) passing some of the apartment buildings where some floors are actually “dorms” for sophomore and later years.
This way, she will get an impression how this “feels” to her, vs. other campuses she’s seen.
PS: You’re saying you’re taking the train from NJ? So if she’s interested enough, she could decide it’s worth taking the train herself for the full info session/tour some other week/month?
And/or, if very interested, she could arrange permission with her high school to sample one or two days of real classes of her choice at Barnard while college is in session.