Since the attached reflector and the rest of the aircraft are at the same position and velocity, they form a single target. There will be no second blip. They're one and the same.
When a target has moving parts, like exposed engine blades, they distort the signal, producing a radar signature. This signature will be obscured by the much stronger reflection from the Luneberg lens.
This is a small side effect of the attached reflector's purpose: to allow allies to relatively easily track and target the aircraft. However, such a handicap also keeps them from experiencing stealth as it is and developing tactics to deal with it.
Even when high-res radar imaging is used, spatial separation alone is generally not precise enough to separate the signatures of other airframe parts from the removable reflector on its own. Engine signatures (JEM) can be separated through doppler shift. When the engines are concealed, there's not much in the way of distinguishing the airframe's static signature from the reflector's.
For more technical information, look into the subject of NCTR (non-cooperative target recognition).
Some ACARS papers can be easy to digest.
Pages 177-184 (28-1) make for a good starting points on the basics, 97-110 (8-3) address reflectors and their effect on the radar signature more specifically.