When I try on the illusion on an offhand artifact (tested with my Rogue), the illusion doesn’t show up in the transmog UI, but it DOES show up on the character after I apply the mog. Test for yourself and see if it works for you as I describe. Here’s my Rogue to show what I mean:
You can see that the Berserking illusion doesn’t show up in the transmog UI, but does show up on the character after I apply.