I'd only do a 1x 10 i the bike is already a 10 speed cassette, and brifter.

Calculate the Ratios you need I doubt a 12 and a 13 t will be as useful as just one of those .

As a single chairing , using a double crank [withoutchanging the bB spindle ]
neither chainring will be in a good chainline,
as both are offset on either side of it.

instead consider just closing up the difference between the 2 chainrings, leave it as a double .

something like a 42/48t perhaps? or the Paris Roubaix folks race on a 46/53
as the course route is pretty flat , and fast is what keeps you at the front.