It seems to me that the bath shouldn't be the issue. The issue would be the fleas. Why not skip the bath and put your cat on flea prevention?
Advantage and Frontline are great products for flea and tick prevention. Keeping your cat on one of these products will eliminate your flea problems.
There is also a product that you can purchase from your vet clinic called Capstar. It comes in pill form and when administered it will kill the fleas within 30 minutes. It will NOT prevent fleas from coming back so this is not a cure all. Capstar must be used in conjunction with a flea and tick preventative.
Also remember that when your pet has fleas it will absolutely have tapeworms. Tapeworms come from when a pet ingests a flea. That issue will have to be addressed and treated. Be sure to treat your dog for tapeworms also. All it takes is for your dog to swallow one single flea to have tapes. Deworm him to be on the safe side.
Bathing a cat is definitely not a thing of the past. If your cat is up to date on vaccinations then I would recommend a bath from a groomer. Let them handle your baby if you feel that your cat needs bathing. It is relatively inexpensive compared to the trouble that you go through attempting an in home bath.
