Don't waste time talking technology/speed/life cycle with this guy .. he doesn't care. He thinks that you are using the project as an excuse to do PC upgrades and increase your fees.
I would change the subject from the age/speed of the computers to the time it takes people to do the work. Install the software on one of the dogs and test it. If the software performs acceptably, leave it alone. But if it doesn't, the argument is about how much the people cost and their productivity.
Anecdotal info ... Just this week we were exploring memory upgrades for our accounting staff. For recent PCs (18 months old), we got 2GB of RAM for $32. For some older PCs (3+ years old), we could only get 512MG sticks at $80, so 1.5GB would cost $240.