The American stealth that you are referring to here is based on the older F-117. Even if the Rapier battery could have ID'd it (without it squawking IFF codes), congratulations...an ultra-modern missile managed to pick up a 2nd generation stealth design (SR-71 being the first) that has been in service for almost thirty years and is being considered for retirement!
Even if the Typhoon is more maneuverable than the F-22 (which no one knows for certain, but I doubt it thanks to the F-22's thrust vectoring), the F-22 would still be able to pick up a Typhoon far further away than a Typhoon could pick up an F-22. At this point in time, where missiles can kill things BVR reliably, maneuverability is a luxury. Stealth is the key in the modern world. That's why the designs for the latest front line aircraft are stealth-based (e.g. B-2 (last bomber designed), F-22, and F-35).
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