There are many ways to express this meaning, e.g.,:
You will/shall pay a part of the payment plus two months in advance.
You will be paying a part of the payment plus two months in advance.
You would pay a part of the payment plus two months in advance.
You would be paying a part of the payment plus two months in advance.
(The word "plus" could be replaced by "and" or "along with" without changing the meaning.)
It's often the case that a more indirect construction sounds more polite, which would be appropriate when talking to a customer. Here, the third example sounds more polite than the first because it's expressed as a hypothetical rather than a command, which suits the fact that the customer has not decided yet. That's probably the option that I'd choose with the information you've given.
I'm not sure the progressive tense ("will be paying") rather than the simple future ("will pay") gains you anything; the longer verb phrase softens the requirement, but it also introduces the ambiguity of whether the customer will be paying in multiple installments or a lump sum. It can also sound unusual—in American English—to use the progressive tense repeatedly (e.g., "I am doing this; you are doing that; I will be seeing about this; you would be paying that" does not sound natural).
You could also express the outcome in terms of what you will do:
We will/would collect part of the payment plus two months in advance.
Finally, it's not always true that a shorter expression is impolite. For example, you could also use the present tense to succinctly describe the outcome:
If these conditions are agreeable, I propose we move forward with the following steps: (1) You pay part of the payment plus two months in advance; (2) ...
However, these points all involve nuances; none of the statements is incorrect or particularly rude. You might notify the customer, if you haven't already, that you're not a native English speaker and that you're not trying to be brusque but rather clear and succinct.