What you need is a good scraper (in German: Kratzer) made from a quality file. It is and has always been a very popular and efficient tool used by professional accordion tuners.
Scrapers are the one tool I haven't tried, but will. Do you simply sharpen the business end, or do you sharpen then create a burr, like with a wood scraper?
Like Diatonix, I use a scraper made from an old needle file. I have ground away the old teeth for a few centimetres, so that the sides of the reed plate slot are not damaged when working on the underneath reeds. Then I grind away the original taper on the last centimetre of the tip to create a cylindrical cross-section about 1 to 1.5 mm diameter. If you look carefully at Diatonix's photos you can see this cylindrical zone on the tip.
Finally, I grind the tip at right-angles to create a flat circular end with a sharp 90° edge all round. No need to burr the edge over as you will be using the scraper primarily with a push action. I do all this grinding and shaping by hand using an old carborundum oilstone and a flat diamond-coated steel plate.
Initially, I made the mistake of simply leaving the original taper of the file in place until it was pointed out to me that would create a scraping edge of >90° and therefore less efficient as a result (thanks for the tip Theo!). Neither do you want to have a needle-sharp point on the tip of your scraper which simply digs into the reed tongue.
In use, the scraper needs to be frequently touched up on the oilstone or diamond plate to maintain the little circular tip with the sharp 90° edge.