If touching up with paint/varnish is not an option....
Make a cabinet/card scraper from an old handsaw .
Use an angle grinder and buy a 1mm burn your clothes disc , about 2 quid .
to cut the saw up .
Buy a single cut farmers own file (a fine cutting one) for a bit less than a tenner .
This file only has grooves going in one direction, not crossing over .
Get a perfectly square block of wood so you can register the file against it and the edge of the saw
at a perfect 90 degrees ...
I would be using the the factory edge back of the saw for the cutting edge if it has sharp corners allready....
otherwise not.
and just running the file along it the sharp edge to get a flat, even file mark along it .
Then repeat this process with the block on a (preferably) flattish oilstone or a pack of gradually fine wet n dry sandpaper . I'd try get a stone
You need a DEAD flat surface for the paper
Marble or cut stone or something ...wet the back o the paper for suction .
and clip the corners off the scraper so as not to dig into the paper .
Then you need to draw a small burr ...do this with a hss drill that is not scuffed
bit like around an 8 mm bit with light pressure .
Thats about the cheapest way I can think of ...
The perfectly square block is a real help if you don't have the muscle memory from practice .
Google how to sharpen a card scraper .
Good luck