I want apple to zero my balance so I can change my region

Here are two ways you can remove all your Apple ID (Apple Account) balance:

1) Spend it:

If your store credit is less than the cost of a single item on the iTunes Store, App Store, or iBooks Store, and you still have a valid payment method for that country, you can add the payment method to your account [ how to add a card --> https://support.apple.com/HT201266 ]. When you buy an item, it will use the remaining store credit and the rest will be charged to the payment method.

2) Have Apple keep it

"If you still can't change your country or region, or you have less store credit than the cost of a single item, contact Apple Support." Apple will remove the balance and keep it.

Click here --> https://support.apple.com/choose-country-region/media-services Select your country, then a product such as billing. If you do not see your country, see if there is an "other" listing for that region. If you don't see an option that handles your issue then keep experimenting with selections until you reach one that gets you a chat session or a telephone call and get the representative to redirect you.

Or call Apple Support --> https://support.apple.com/HT201232

References:

Click on this link to read full information about requirements for changing store country:

Change your Apple ID country or region --> https://support.apple.com/HT204447

You do it using one of the two methods I outlined above. You either buy something and Apple will automatically change any remainder to a card you have on the account, or you contact Apple using one of the methods I provided. You then ask them to remove the balance fro your account. You have to do this. I do not work for Apple. All I can do is tell you things you have to do.

I menioned two ways in my second method. Which one did you use? Did you try calling the telephone number? With the first way to contact Apple you have to just try different options until you get a chat request. There isn't a specific one for asking for removal of account balance.

I have never, ever heard of anybody being asked for a serial number when doing this.

You do not need a serial number to change region. Serial numbers apply to hardware, your region change is on your Apple account.