Personally, I would take ANY payment a customer wants to make!  I do understand your expectation of having payment in full.  The best way to encourage this is with a business practice not a system feature.

The business practice would involve a notice on your invoice which stipulates a service charge for partial payments (you can decide how much that charge should be).  If a customer pays a partial amount, you could apply the service charge and send them a statement.  The customer will catch on quickly.  Be sure to warn them in advance to meet full disclosure practices and make sure this is lawful in your part of the country.

Best,

Kevin