Just let me know when the payment HAS BEEN MADE and I’ll keep an eye out for it.
I know that I should drop "will" after "when", but why Future II tense was used?
In some languages (e.g. German), the term "future II" is used to designate the future perfect. The term "future perfect" (or "future II") refers to constructions such as "will"/"shall" followed by a perfect - e.g. "I will have gone".
Also in some languages, the future perfect is used in phrases like *"When I will have paid" and *"When the payment will have been made". The future perfect is not correct here in English. Instead we use the present perfect: "When I have paid", "When the payment has been made".
You conceptualise this as using the future perfect but dropping the "will" after "when". This is not helpful in my view. It is better simply to say that English uses the present perfect, not the future perfect, in this scenario.
(Similarly, we use the simple present where some languages might use the future. For example, "When I'm 90 I will be old" (not *"When I will be 90 I will be old"), or "Let me know when you arrive". Now, you can also say "Let me know when you will arrive", but it has a different meaning. "Let me know when you arrive" means "let me know [something, possibly the fact of your arrival, possibly some other information] later, once you've arrived", whereas "let me know when you will arrive" means "let me know [now or whenever] what time you expect to arrive [before you've actually done so]".)
In any case, we use the perfect in "Just let me know when the payment has been made and I’ll keep an eye out for it" to represent the fact that the payment will have been completed by the time you let me know that it's been paid.
The tenses are correct (although it would also be possible to say "Let me know when the payment is made" - but you couldn't say "Let me know when the payment was made" because that would imply that the payment had already been made at the time of the conversation, rather than the clerk telling the customer to let them know once they'd made payment).