Household paint strippers are too aggressive, and will dissolve your gelcoat. Paint strippers that are designed specifically for fiberglass seem to work as well as other types, without harming the boat. If you used it on your keel without damage, it might be that your keel is lead or iron, and has no gelcoat on it. Klean Strip (kleanstrip dot com) manufactures a chemical paint stripper for fiberglass that I have used, and found it very satisfactory, and it's much less expensive than some marine strippers.
Stripping years of accumulated antifouling paint off the bottom is hard, unpleasant work, and the only easy way to do it is if you hire it done, but chemical strippers will get the job done, if you're patient and determined. It helped me stick with it to think about how fast the boat would be when the bottom was smooth and clean (and I wasn't disappointed).