For this type of repair I would normally trim the damaged area, fashion and glue in a wooden plug with hide glue, then inpaint and finish the damaged area. When time or budget constraints are a limiting factor this is a technique I use.
Epoxy putty sticks consist of a resin and hardener in one stick you just cut off a piece and knead it together to blend the two parts, it then cures in less than ten minuets. It is available from several sources and in many hardware stores. It is available in several colors. The manufacturers claim it to be permanent. I have not used it long enough to know how long it lasts. It seems to me that in order to be permanent the coefficient of thermal expansion would have to be the same. So far I have never had it fail.
Once the putty is kneaded it is pressed into the defect and formed to the correct shape but left proud of the surface. It can be formed with your fingers or tools. I always use a fill that is lighter than the color of the base surface.
Once it sets but before it is totally cured, trim it to match the surface. It can be trimmed with chisels, files, floats or sand paper, even a small plane.
If you trim it to soon it has a tendency to chip out of the repair, if you wait to long it becomes quit hard.
Next I apply a coat of garnet shellac. It is always easier to match color after a coat of finish is applied to the repair and the surrounding area.
I begin to build up the color with dye in shellac. Start light and build up to the right color, it is easier to make it darker than to make it lighter.
First coat of color.
Here the color is about right but the sheen is off. You can see that I have scratched the grain into the fill with a dry point tool and colored it dark.
A little bit of toner and a top coat blends it right in.