Yes, you can use the word "amount" when referring to money. Both "sum" and "amount" are commonly used to describe a quantity of money. So, "the total amount" and "the total sum" are both grammatically correct and interchangeable in most contexts. However, "sum" may be considered slightly more formal than "amount." In general, it's a matter of personal preference or style guide as to which word to use. Some writers or speakers may use one term more frequently than the other, but both are acceptable in standard English.

You are correct that in this case "sum" and "amount" can be used interchangeably. "Sum" also means "add": "sum (up) the numbers" = "add the numbers". Therefore the "sum" is everything added up. So sometimes people use the word "sum" to include this extra connotation of everything being added up. "Sum" is also similar to "balance" in this way. It suggests that all the numbers have been accounted for and the resulting number is final.