OK, a plan is forming.
Use a multi-tool to cut-out most of the rotten timber, with a bevel on
the cut edge so that the joint will be slightly scarfed
Use Ronseal Wet Rot Wood Hardener to stabilise any dodgy wood that I
can't remove.
Fix some treated battens up the back of the good board to support the
new timber.
I'm tempted to use a PU adhesive in the joint - is Everbuild Lumberjack
Polyurethane Wood Glue any good?
Does this seem like a reasonable plan?