Thanks a lot for all the great tips!

I ran the hot water tap closest to the pipe for awhile to get it nice and hot. I then turned off the main water line and drained the pipes (to get the layer of room temperature water out of the pipes). I then turned it pack on, theoretically filling the pipe with fresh hot water.

It worked, I was able to feel the temperature difference in the two pipes.

Now I just have to install the valve, hang the sheetrock, buy the tile....