Exterior Door Outside Handle - Smaller than new one so holes do not line up

Our original (40 years old) exterior door handle broke - it is the handle on the outside of the door (where it has two holes - one for the lever to connect to the interior knob, and the other to hold the bottom of it in place. Like this one:


So we bought a Schalge lockset with the assumption these were all a standard size. Our Schlage is now longer than our original, and even with the adjustments you can make, does not fit the bottom hole. If we were to install it, there would be a small hole exposed from the outside in.

We plan to eventually just get a whole new exterior door, but have been dragging our feet on it, so we want to keep the Schlage set.

Any ideas on a solution? Is there some sort of steel plate you can buy for this purpose that will cover the entire lockset or something? The door is wood, and while I could use wood plugs and attempt to stain them (to match) I worry it would look very poor, especially considering I need two different stains since the exterior has aged much more than the interior side.