Technically, all Titan IIIE and Titan IV rockets lifted off with only the solid rockets burning - the center core ignited in the air. Photos which appear to show exhaust coming from the core vehicle are misleading; those are really just parts of the solid plumes getting sucked up into the liquid engines and swirling about. So if you REALLY want accuracy, you'd lift off under motors in the strap-ons, jettison the strap-ons, and light the core in the air. Not a trivial task, to say the least.

As for putting motors in the plastic engine bells on the Estes kit, they are too small for 18mm motors. There IS room for dual 18mm motors in the main tube, but the plastic engine parts would probably have to be left out for flight.