What I would do is remove the rotten one by cutting it in half so you can get both parts out. Then use the proper sized hole saw or forstner bit to open the hole in the bottom rail all the way through the lower rail.
Added Info regarding the lower hole drilling.
- Create a starting guide by using the hole saw or forstner bit to drill through a thin piece of stock like 1/4" or 3/8" plywood.
- Position the starting guide piece with its cutout directly over the hole on the top of the lower rail.
- Position a second piece of wood such as a 3/4" pine board under the lower rail as a backer board to help prevent breakout on the lower rail bottom.
- Clamp the starting guide and the lower backer board to the lower rail using some C-Clamps or Squeeze Clamps.
- Drill out the hole in the lower rail slightly into the backer piece.
This method really helps to keep the saw/drill bit aligned properly when drilling. It can also help a lot when/if there is a desire to enlarge the holes some to trim out decay in the old hole. Repeat a similar procedure if needing to enlarge the hole slightly up into the upper railing.
- - End of Added Info - -
Cut your replacement piece to the length needed from the bottom of the lower rail up to the inside of the pocket in the underside of the upper rail.
You should then be able to lower the new piece through the top of the lower rail just enough so that you can bring it under the upper rail. Then raise the new part into the pocket in the upper rail.
Secure the new part by nailing or screwing in from the sides of the lower rail into the lower end of the new part. Drilling clearance and pilot holes is advisable, especially if using screws.