It works perfectly.
The task scheduler runs script #1 every two hours using wperl.exe. Script #1 checks to see a backup is due, and if so, it calls exec() to run script #2 using perl.exe. Script #2 interacts with me using its command prompt window and does the backup.
P.S. The reason I had to use exec() rather than system() was so that a new process would get created. At least, that's my guess, since when I tried to use system(), the second script appeared to run under the same Perl executable that the first script was running under.