You can set a Bcc header by using the 'Bcc' key in the call to MIME::Lite->new. And you can give it multiple addresses by passing a reference to an array of addresses. You might not be able to send 4,000 messages in one go, but you'll certainly be able to send them at a better rate than your current one at a time.

During this period when someone visits our site, a visitor will occasionally get an internal server error which I'm think is related somehow to the mail/sendmail program

Maybe you should consider using a mail server that isn't also your web server.

Your require "Lite.pm" is rather non-standard. You should really replace that with use MIME::Lite. There are a few other unusual pieces of code in there, like the way you unnecessarily read the whole address file into memory. The whole program would benefit from being reviewed by a more expert Perl programmer.

And finally, more and more people are treating pure-HTML email as spam. For example, I never get to see any mail message that doesn't have a plain text version attached. If your message isn't spam then you might consider making it look less like spam.