Mega-Gangar/Smart-Inventory ? reverse-engineered prompt

Build me a Flutter app for a small shop that works mostly offline and lets the owner manage inventory and billing in one place.

I want a clean login and registration flow with Firebase Auth and Google sign in, then a simple Material 3 dashboard where I can add products, edit stock, and see what is available right away. The billing part should let me create a new sale, pick items, change quantities, and automatically calculate the total as I go. If there is not enough stock, it should stop the sale and show a helpful message. Please save all data locally with SQLite so it stays fast even without internet, and keep sales history and profit info in the app too.

I also want a print friendly receipt or bill layout, a nice modern UI, and Android friendly app icons. If you need to check current Flutter or Firebase docs online, go ahead and do that.