I've used a steam box for a similar issue with excellent results. The box was roughly 5" x 5" pine and long enough to steam the entire barrel channel (and span our kitchen stove). It was open on the bottom for the steam with holes at each end for the stock. I used bicycle inner tube over the end holes, slit to provide somewhat of a seal. Underneath, I used a long gardening tray to hold the boiling water (the ends of the box were tall enough to allow for the gardening tray to just fit underneath). Rather than steam the stock with the barrel in, I made a wood "half" barrel, profiled like my actual barrel, that came up only to the top of the barrel channel. When wrapped with bicycle inner tube, this helped pull the stock tight to the sides of the "half" barrel. I let it dry for several days.
One important tip: Make sure you overlap the inner tube wrapping. Wrapping with spaces, like a barber pole or candy cane, will result in a wavy forestock, closing the gaps only where wrapped. Guess how I know that........
Bill