Straight screen pin

I need to replace several plunger/spring pins on window screens on my house that have fallen out over the years. I have tried most stores to see what is available and all the options are way too big. I have done my best to measure the pins themselves but can't find a replacement.

The diameter is 3mm x 2 inches long (a #0 screwdriver fits PERFECTLY) in the screen hole. The issue though is all the plungers I have found have a bulb or 'widening' on one end and that doesn't work for the screen and insert hole (in the actual window frame). I need a straight pin of the above measurements with a removable cap to be placed on the inside of the screen (once pin is inserted through the screen hole and into the hole in the window frame. Do you have anything like that? Or any suggestions? Thank you!!

Top to bottom as follows:

All pictures should be turned 90degrees clockwise.

1) existing pin, with cap on out of insertion pulled out

2) pin and cap as is

3) when cap and pin in correct placement

4) hole through screen frame without pin

5) window frame hole where pin secures (notice the inside of hole is NOT completely oval- the outer edge of hole has a flat side)

Any pins I have bought so far have a "bulb" on the outer edge - which doesn't fit through screen hole or frame hole. I hope these pictures help. Thanks!