I didn’t have a circular saw. I used brace and bits and a portable jigsaw to cut the rough shape of handle. After that, I simply used a small dovetail saw. I marked the cutting line using a marking gauge, adjusting the tool to get an exactly centred line. I did saw very slowly and very carefully to maintain the saw blade in centered position and equidistance from both sides of the handle. It took me a while, but it did work. I passed very carefully a sheet of 140-grit sandpaper to clean the walls of the kerf, helping the sandpaper with a 0,5 mm-thick metal sheet, to maintain sandpaper flat and straight. After that, saw plate got into the handle with the adequate grade of friction and completely straight. Since the thickness of the saw plate was 1 mm, I used a saw with the more similar size of kerf.
The funny thing is this: I found that dovetail saw in a flea market and paid 0,50 euro for it. I restored and sharpened it for ripcutting (it has 14 tpi). It’s a very old William Hall saw, but it’s one of my best saws now. It never fails.
I leave here a pair of photos of the process.