Break-up vs. Break-down

In the example sentence you have, you need a noun, not a verb (the definitions you linked to are for the verb). You also need an article (whether it's "a" or "the" depends on context).

Thus you need to look at how these words are used as nouns.

"Breakup" as a noun refers to an event where a group, organization or pair is dissolved or otherwise stops being a whole, such as the breakup of the Soviet Union (i.e. into Russian and many other countries) or the end of a romantic relationship. It's not used for dividing things into categories (which is purely conceptual) and it wouldn't be used in your example sentence.

"Breakdown" as a noun refers to (in addition to its other definitions):

division into categories · a statistical breakdown of data : classification; also : an account analyzed into categories · a budget breakdown
_{Merriam-Webster}

This is exactly the meaning you're looking for with your example.

In the context of you question, of requesting an itemized list of an invoice, it would be more appropriate to ask for a "break down" of an invoice.

Whilst both are used to refer to separating something into its individual components, "break down" can mean to go into greater detail, or to examine the individual components, whereas "break up" is usually more destructive

Break down

Break up