Hire two children to hold the bases for you, shouldn't cost more than 50p an hour 
I'd suggest Command Strips. They are two strips of velcro with a gel type glue on the back which can be removed quite easily without leaving a mark on the surface they are attached to.
I use the bigger ones for hanging pictures/paintings in the house* as well as large framed photographs for exhibition. Works a treat.
Here is a link to them on the 3M website:
http://www.3mdirect.co.uk/command-strip-products.html
Never game on desk you're not prepared to get scratched, coffee stained, drill holes in etc
Not helpful? OK, try these ...
1. Don't use your desk at all. Mount the joystick to the arms of your chair instead. Much more comfortable. Your back will thank you later.
2. Clear your desk and place a large flat piece of glass cut to the surface dimensions of your desk. Mount some suckers to the base of your joystick and stick them on the glass surface. This has the added benefit of protecting the whole surface of that lovely pine desk.