Well, I'm not saying it is a Paramount, per se. It isn't badged "Paramount", as later Paramount Tandems were, but rather just as "Tandem". I'd sooner say that it is, essentially, the forerunner to the Paramount Tandem line.
It seems to be constructed much the same as the later Paramount Tandems (fillet brazed, chromoly tubing), and the fork is a Paramount design; the same basic fork is shown the 1938 catalog as an available part, and is distict from the Superior or New World forks. The AS&CO chainring appears in '38 as standard on the Superior, and available for Paramount touring models. Also, from what I've found the X003 format serial number seems to fit in with serials of roughly that same time, but it does seem more likely that this serial is for a unique series, and not intended to fall into the mainstream production numbers. You're right, it does have mounts for chainguards (but may have never actually had them) and for front and rear fenders.
At this point, my best guess is similar to your suggestion - that it was a custom order, for a racing tandem built to Paramount/Superior specs rather than, say, Town & Country. It could be that because of orders like this that Schwinn eventually decided to add a tandem to the Paramount line. But doesn't lead me any closer to a build date.