in the past I put something up on the wiki about this, and today i actually went and did it.
DMV told me to go fuck myself even though I SHOWED them the regulation from their own employer stating I don't need a title from a non-title state, just a Bill of Sale.
A shitload of other title places told me the same.
THEN I found a place in my town that said "Oh, there's an exception for that."
New title, tax, plates and everything plus their $25 fee was $153. Expensive, but cheaper than a bunch of tickets.
If you can't find a titling place or dmv that will just register the thing, then you CAN try to register it as a dirtbike, "sell it", get it titlted somewhere else as a moped, then "buy it back" and re-title it. It's a bit much, but to get legal it's worth it.
So, there you go, you CAN title in Illinois as long as you have a BOS from a non-titling state (MO, and IN, I think).