2002 MCS: I replaced the key shell because the logo had fallen off and the key itself was a scratched up mess.

Now, the Door lock/unlock works, but the engine won't start. All the lights come on, but it won't start. I've tried all the voodoo combinations of button pushes but nothing

Interestingly, if I start the car with the Valet key, remove it, and put the remote key in, it DOES start. But, leave it out for a while, and it goes back to not starting again.

Any ideas?

 

Sounds like you didn't transfer the chip the ECU uses to verify you're using a valid key. Try holding the old key next to the ignition while starting it with the new key to verify.

 

That's exactly what happened to me... And what I didn't do. In fact, I almost threw away the old shell after transferring the electronics and actual "key". Only much later (and a lot of gut wrenching and head scratching) found the RFID chip in the back of the old shell. Rather than transferring the chip I just reused the old shell back with the new shell front, with its MINI emblem. And now she cranks!