There are no 'individual processes' on Windows when using Apache, there is only one Apache worker process. Also, there is no daemon mode on Windows either. Anyway, what it all means is that both Django instances do run in the one process, albeit in different sub interpreters. So yes, to cause code for one Django site to be reloaded will have impacts on the other site because of them being in the same process.
If one was on a UNIX system one could send a kill signal to the worker/daemon processes and they would restart without restart whole of Apache. On UNIX this does not overly cause problems even when multiple sites in same process as the listener socket for port on which HTTP requests are accepted is maintained open at all times, so subsequent requests which arrive during the restart just queue up and will be handled once worker/daemon processes are running again.
On Windows as you rightly pick up the whole of Apache has to be restarted. This means that for ever so small a time that listener socket will be closed off and there will be a small window where requests will get a connection failed. How long a window this is you will really need to do some testing on. Usually it isn't a huge problem as is short enough that low probability that request will hit at that precise moment. In other words, you may be unduly worrying if you are only talking about a staging environment given that wouldn't expect to restart very often. If you were trying to run a development site on same Apache instance then that would be a problem.
That all said, if you want the staging instance to be as close as possible to production then would still need to be running Apache so multiple Apache instances on different ports would be only logical solution. You could run Django on top of CherryPy WSGI server or Paste WSGI server and proxy to it, but then it is a different hosting system and will behave differently to the extent that you may not pick up issues which would ultimately occur when on production setup.
All up, I would suggest you actually do some tests where you run benchmarks against Apache and perform a restart at same time and see how many requests fail as a result. This will help you understanding whether it is the big problem you think it is.