Don't let the rude lady (and I use that term loosely) win! If you want to work out at the gym, then go. Fat is simply a state, and you can change it. She will probably be rude and inconsiderate for the rest of her sorry life.

But, if you feel more comfortable working out at home for now, then do that. No point torturing yourself. The main thing is to get some exercise - it doesn't matter where.

Stick with your program, and take care of yourself, because you deserve it. You will start to feel better when you see some results, and I bet that starts a whole positive feedback loop - you'll feel better, treat yourself better, look better, and so on. Then you can march back into the gym and make the rude lady eat her words.