The tense is fine, but the sentence should really be:
We will pay and sign the booking form next week.
Or if the meaning really is that the form will be signed a week after payment
We will pay Thursday and sign the booking form a week later.
Just let me know when the payment HAS BEEN MADE and I’ll keep an eye out for it.
"has been made" is a present perfect form, it is used when the speaker is looking back at an action that was completed in the past, and remains in the completed state. In this case, at the time the payer "lets me know" the payment has already been completed, and is stays completed. That time is the viewpoint for this sentence. In casual speech it might be shortened to
Just let me know when the payment's been made.
or an active voice could be used
Just let me know when you've made the payment, and ...
Just let me know when you've paid.