[Wavery]

If it were me, I'd take out as little of the original wood as possible. I'd carefully (and squarely) remove all of the wood rot then scarf in a new piece of wood. I'd use a "Multi-tool" or a 4" circular saw (or a combination) and make the cut as square and straight as possible.

See pic and youtube "How to scarf wood" or something. It isn't hard. I had to repair a wood mast on my sailboat one time in Africa. I learned how to scarf from an African shipwright. It was actually fun and when I sold the boat (70,000 miles later) the mast was still perfect. I even sailed through a cyclone with it between New Zealand and Australia in 1995.