Good luck with her move-in. Barnard (like all colleges) have their procedures proofed over many years. I believe they went as far as partially closing the subway station for much of the day, to avoid people coming up the stairs THAT side of Broadway when the sidewalk is already full with parents and students.
Are you planning to rent a car to do your shopping in the ‘burbs, or are you planning to do your shopping trips in cabs?
If you’re staging everything at your hotel, then you could just do a few trips during the remainder of the day and wouldn’t be limited to whatever time slot.
In prior years you would arrive a few minutes early and pull up/double-park along Broadway. You’d look for an empty square painted on the sidewalk. While your daughter disappears to get in line for registration and get her ID and room key, the parents unload the car onto the square. One parent babysits the goods, while the other goes around the block to a parking garage (figure out where those are ahead of time) - one is right on Claremont.
It might work out that daughter comes back by the time driver walks back from parking garage.
Next step is for daughter to potentially take a bag and try to be first in the room to have first claim on the preferred bed 
and put her bag on THAT.
In the meantime one parent tries to secure a rolling moving-“bin” and the parents start loading that up. When daughter returns from mad dash to room and back, she and a helper will push the bin to the room where she can unload.
Dorms are small and there really isn’t much another adult can do since furniture, desk, chairs are already standing and the kids will want to organize their own space anyway.
After max two trips all will be in the room. If you can’t fit it in two moving bins, you can’t fit it in her dorm!
At this point ask your daughter what she prefers: maybe give her until afternoon to explore/socialize/unpack a bit — it’s not a race. They’ll have plenty of downtime during orientation to get their space set up.
Maybe by the afternoon she is up for an early dinner, or might have realized what else she might need, or what might not fit and can be returned, requiring one final shopping excursion.
Parents could return to Hotel, attend whatever program that Barnard might be hosting, or walk to Riverside Park, Grant’s Tomb, etc. and wait for a txt.
. I did immediately order headlamp, flashlight, and a small lantern to pack last night. Thinking I should buy a few gallons of water when I’m there despite our family being a huge fan of personal water bottle refills. And maybe a tiny portion of things like crackers; tuna, beef jerky m and peanut butter since they don’t require any microwaving.