RE: Is there ANYWAY to choose how the payment is applied to the classes? I do not like the way QB does it by percentages. Sometimes that's fine, but not always.

No, there isn't.

The balance sheet won't use the classes on the detail lines of the - in most cases - because they to use balance sheet accounts. It uses the class at the top of the invoice, the class associated with the Accounts Receivable line item.

Notably, generally class tracking was originally implemented for your P&L, for income and expenses by class. Later, Intuit added the class field to the overall transaction, like the AR account for an invoice, or the bank account for a check, and that line item, like all the others on a transaction, can have one class associated with it. Never more than one.  A balance sheet by class will never see the classes you've entered on the detail lines of a transaction, unless they themselves use balance sheet accounts - and then it might, but I'm not quite sure.