I think you might get the best results using fork: the parent process spawns a child process (and has the child's pid); the child does an "exec ..." to run the external program in question. This makes sure that when the external program is running, it has the same pid as the child process that started it.

The parent could check the age of the file, or check the size (with the "-s" operator -- see the -X section of the perlfunc man page), sleeping for maybe 3 or 5 seconds between checks. If the file hasn't grown since the last check, kill the child pid and start a new child. Continue doing that as long as you want.

(Update: added link to docs for the "kill" function, to clarify that this is a built-in.)