I tend to go to their desk (if in same office) or pick up the phone and speak to them, so I can find out what the issue really is, and often find out they aren't being arsey at all.
Also, in that string of emails, your initial email is the first to be arsey...
Here's some more arsiness for you...are you planning to tell us which of the many strings of emails in this thread you are referring to?
Your's.
Being arsey != being assertive.
Don't demand the payments, explain that they are overdue, and the impact this will have (politely.) Offer assistance, even if you and they are fully aware you are not in a position to do so.
What you're doing at the moment is generating conflict, not resolution.
that sort of reply is an example of exactly what NOT to do, if he has already had cause to complain he now as a arsey email from you...
a better reply would be
"sorry to trouble you again, be we need to get your PC swapped out today to keep the project on time, if you could let me know a convient time to swap the machine over I could be much appreciated"..
you got a good reply from him, however it could have very easily gone horribly wrong... what if he had taken offence and asked you to leave?
lawyers are be total arses I did a lot of installs for them...
I once signed an introduction email:
"Any questions please do not contact me."
I missed out 'hesitate to', lucky for me the customer saw the funny side of it.