What you want is called a soldering blanket or a flame arrester. It
has two brass grommet in it so you tack it in place in front of the
wood. In the old days it was made out of asbestos. Today it's made out
of the kind of stuff they use on the space shuttle. You'll find it at
Home Depot near the solder in the plumbing department. Cost is about
$15.

I've done soldering in even tighter spots. But then, I have a few of
those blankets.