I second the recommendation to apply to Barnard during RD. Barnard is considered one of the 4 undergrad schools within Columbia. The others being Columbia College, SEAS and GS. Students among these 4 schools are intermingled in the undergrad course offerings. The schools have differing graduation requirements.
Although Barnard has its own trustees and management, it is closely linked to Columbia. The diploma I believe says “Columbia University Barnard College” or something like that. Students at CC/SEAS can take Barnard classes and vice versa. Some students actually prefer Barnard because the core requirements are more relaxed.