Question: How would you suggest getting past the dreaded “we do not accept unsolicited material” notice?

It’s really simple: you send a query letter.

Companies DO NOT want to see your screenplay; that is the “unsolicited material” they don’t accept. They’ll accept query letters all day long.

Do your due diligence: find out who you contact, and send your query letter specifically to that person (or department). That’s the professional way to get your foot in the door and, ultimately, to get the company to ask to see your screenplay.