Rav Yehuda Henkin, the rabbinic supervisor of this site, does permit the use of the contraceptive sponge since it does not prevent sperm from entering the vaginal canal (as a condom does), and does not interfere with normal relations (although it is not inserted as deeply in the vagina as a diaphragm).
As you have already encountered, this second point is subject to halachic dispute. Some halachic decisors are strict about barrier methods that take up a greater presence in the vaginal canal.
While the sponge is not a highly effective method of contraception (and is even less effective for a woman who already has children), if you are fully nursing, and your baby is under 6 months old, and you have not yet resumed menstruation, you are naturally protected against pregnancy to a reasonable degree. See our article on the Lactational Amenorrhea Method for more details. Using the sponge, or spermicide, (especially VCF which is somewhat more effective than other types of spermicide) as a backup method offers an additional level of protection.
The diaphragm can be placed up to one hour in advance of relations, allowing for some degree of spontaneity. The sponge and other spermicides also need to be inserted in advance of relations, so it is unclear why they are a better option from that point of view. The combination of diaphragm and spermicide is significantly more effective than the sponge or spermicide alone.
Please feel free to get back to us with any further questions.
B’Hatzlacha!