How can I reduce the thickness of a board except in one narrow area?
If I had to do this, I'd just plane down the entire boards on a thickness planer, cut a 3/8&quot; x 1&quot; x 7' board and glue it back in the desired place.  If you don't have a thickness planer, maybe a friend does or maybe a mill shop in your area would do it very inexpensively. A 18 ga nailer with 3/4 nails could be used to hold the 1x7 in place while the glue dried, or even a bunch of small screws.
I know I'm not answering the question you asked, just suggesting an alternate approach.