Ok, it looks like these places are open on Saturdays:
Empire Budokai
160 W. 78th Street, New York, NY 10024
Sensei: Robin M. Rosenthal
212-628-2908
Schedule: Tu 6-8 pm, Sat 10am-12pm
Jamie Towers Judo Club
630 Pugsley Avenue, Bronx, NY 10473
Sensei: Ralph Reyes
718-931-5132
Schedule: M/W 7:30-9pm; Sat 11:30am-1pm
Japan Judo Karate Inc.
6320 Bay Parkway, Brooklyn, NY 11204-3235
Sensei: Kiyoshi Shiina
718-331-6406
Schedule: M/W 7:30-9pm, Tu/Thu 5:30-9:30, Sat 10-1:30
Kokushi Budo Institute of NY, Inc.
331 Riverside Drive, New York, NY 10025
Sensei: N. Higashi
(212) 866-6777
Schedule: M 6:40-7:40, W 6-7pm, F 5:10-6:30pm, Sat 1:10am-2pm
Oishi Judo Club
79 Leonard Street, New York, NY 10013
Sensei: Shiro Oishi
212-966-6850
Schedule: M-F 12-7, Sat 1pm,
P.A.L. Miccio Judo Club
110 West 7th St., Brooklyn, NY 11231
(718) 246-4755
Sensei: Boris Munoz
Schedule: Wed 6-7:30pm; Sat 10-11:30am
Kokushi sounds good, I'm not sure my stamina is up for a 1:10am - 2:00pm judo class but I'll give it a whirl...