Unfortunately Sputnik doesn't seem to want to decay as fast as it should. It took 5124 orbits to come down. However, since this is a static atmospheric model and density can vary by large amounts in the thermosphere, I decided to see by what factor the density would need to increase to bring the simulation back into line with history. If I increase it by a factor of 3.54, then it lasts 1447 orbits.
Running the simulation on the n-prize satellite with the increased atmospheric density brings the lifetime from 165 orbits down to just 47.
In the future the code will have a much better dynamic atmospheric model which should produce better results without the need for a fudge factor, but for now this is the best I can do.
If you're shooting for optical tracking, you may have issues with a sphere that small. Sputnik itself was a larger and highly polished metal sphere, but was only barely visible with the naked eye. Apparently when most people looked for it, they more often than not saw the much larger rocket stage that put it there.
If you can get anything into space with a perigee of at least 500 km, it will stay there for a reasonable time, but that's a tall order without either a very steep (hence inefficient) launch trajectory, or some kind of attitude control and an engine re-light or dedicated apogee kick motor. Much easier is to aim for 250 - 300 km or so and give it all the delta-v you can.
I calculated the following for the 20g 16.5cm sphere:
Sputnik orbit (235km x 959km): 15.7 orbits
Sputnik orbit + atm. fudge factor: 4.7 orbits
250 km orbit: 0.95 orbits
300 km orbit: 2.8 orbits
400 km orbit: 22.3 orbits
500 km orbit: 133.1 orbits
500 km orbit + atm. fudge factor: 37.8 orbits
660 km orbit + atm. fudge factor: 454 orbits
300km x 900km orbit: 54.1 orbits
300km x 900km orbit + atm. fudge factor: 15.5 orbits
250km x 1200km orbit: 32.9 orbits
250km x 1200km orbit + atm. fudge factor: 9.4 orbits
230km x 2000km orbit: 44 orbits
230km x 2000km orbit + atm. fudge factor: 12.6 orbits
One potential trick that might extend orbital lifetime is launching at night, that way your perigee will be over the night side and apogee over the day side. Solar radiation pressure will then act to reduce your apogee, but increase your perigee. Essentially it'd be trying to squash you into a more circular orbit, and since raising the perigee is the best way to extend orbital lifetime, especially of such a low density object, it could be useful.
The software I have CAN include that in the simulation, but I'll try that some other time.