Lamp wire should work okay - we used to do that years ago. What's the difference between installing the doublet and installing a dipole? A 1/2 wave dipole for 20M would be 33-feet long and you can directly feed it with coax (RG58 is small). If you trim its length carefully you won't even need a tuner.
The doublet's feed line is balanced so you simply connect one wire to each of the balanced terminals on the tuner.
The doublet looks like it is so easy to make that I could put it together this weekend, but I'm hoping to put together a fan dipole one of these days soon, too.
Jeremiah 29:11-13 / John 3:16