I would follow the new vet's advice - if he gave you antibiotics to give her, then by all means, please give them to her. The antibiotic is to help fight the infection and help make her better. Cleaning it with peroxide won't hurt either but do give the meds - it's for Lillian's well being.
And I'm so sorry you had the ordeal you did with the other vet
I don't blame you for being ticked off!!! I would have done the same thing. I'm very lucky - my vet is a guy I graduated high school with - we've been friends for 28 years.
Please keep us posted on how your precious baby does
Sending get well wishes and prayers your way.