Snug Harbor Johnny

Try and see if someone with a little lathe know-how can fix your fine machine.  Perhaps there are machinists forums that might lead you to a local resource.  Failing that, a logical examination of the problem might guide you to You Tube videos that can help you proceed.  Heck, I needed to replace garage door springs, and found what I needed to know on the Internet.  Same goes for a variety of mechanical challenges.  Then again, being handy helps a lot too.  Daniel Boone is quoted, "First, be sure you're right.  Then go ahead."

I'd put a backing piece of thin wood on the inside (with a little Saran Wrap between it and the planks so there can be no accidental gluing) held in place with a couple clamps ... use two horizontal strips on the outside to prevent the clamps from denting the outside of the planking.  Then cut a piece of patching plank from the same wood as the planking ... having a card-stock template (made with trial fittings) can help, then by lightly tracing around the card stock with a mechanical pencil will give you a slightly oversized patch that you can shave where needed.
 
  It should be still be sung before gluing.  Use blue 'painters tape' to mask around the edges of the planking already in place to prevent glue slops.  (That can also be done on the inside before step one as an added precaution.)  Sparingly put glue on the edges of the patch and the planks, and that will act as a 'lubricant' to ease the snug patch in place ... use Titebond or equivalent aliphatic resin wood glue.  If needed due to the shape of the area, place Saran wrap over the patch before the glue 'grabs', cover with another piece of thin wood, then clamp as needed - the wood 'outer sandwich' inside and out will prevent denting from the clamps.  Do not use too much clamping pressure, or excess force can 'telegraph' through the protective wood and affect the planking beneath.
 
  Give it a couple hours at least ... half a day is better - what's the rush?  Then undo everything, and you should be able to LIGHTLY sand the area with very fine sandpaper and a little closed-cell foam in back of the sandpaper.  Everything should blend well enough - of course there will be evidence of a join - but repairs were not uncommon on wooden boats.