I guess killing the program closes the file-handle and flushes the buffer.

Yes, but you can flush without closing the file or terminating the program:

use IO::Handle;
... open LOG ...
LOG->autoflush();

# or flush manually as needed:
...
LOG->flush();
[download]

(Note that $| = 1 applies to the currently selected handle, which is not LOG, unless you explicitly select it (by default it's STDOUT).)