Whenever the rocket is in range, we'll have to batch transfer statistics and commands. For this reason, an orbit that tracks the same course on the ground each revolution would be ideal. GPS satellites orbit at an elevation of 11,000 nautical miles and circle the globe in 12 hours. Seeing as we will probably be below 11,000 nautical miles, we can use GPS for macro navigation and positioning data. Micro navigation would likely be done with one or more gyros with course correction done by rudder fins and small retro jets.
Computers are fast, cheap and small.
The only possible problem that we would run into running a solid-state computer in space is shielding. We're probably going to have to sacrifice a bit of weight and get some lead to cover the electronics. Regular satellites use hardened circuitry, but that's going to be a bit expensive for our purposes. We'll also forgo the use of a nuclear reactor for power in favor of some lithium batteries. Reentry should really be just a calculation of when to fire the retro jets to force the orbit to decay.