I EBF for 10 months, watched what I ate, and went to the gym 4 times a week, and didn't lose a single pound until I stopped BFing. I think it was hormonal- for some reason my body was just hanging onto it.
A lack of calories won't affect your supply- baby gets what baby needs. Baby will, however, take the vitamins and nutrients first, so you won't get them. A low fat diet will lower the amount of fat in your milk which is BAD for baby's brain development, so definitely don't do that. Plus while you're still BFing, the amount of fluid your body retains is higher, and your breast tissue weighs a good amount (in case you couldn't tell...), so that isn't going anywhere.
I know it's tough, but this is not the time to try and lose. Your body and your baby are in control. Enjoy your new bundle of joy and realize there is time later to lose weight.