JohnnyMnemonic wrote:Is there any way to remove the core without picking the 4 pins along with the driver pin (if he can even be set that deep)?
No. Picking might still be possible. Be careful not to turn the core too hard, in case it's the spring that's blocking the shear line as Gordon says. If it gets snagged or bent it might be impossible to get it open again. If you're lucky it might be possible to get the driver pin and spring out through the keyway as well, depending on the size and shape of things...
We've all been there.

It's a risk you take when disassembling locks, sometimes disasters happen, by mistake or just bad luck. It could have been worse, you could have deadlocked it completely.
(My only accident so far was when disassembling a cheap Chinese dimple lock and it turned out that it for some reason had a "ribbed" core with deep groves all around which immediately trapped the driver pins. Lesson learned: Always use a shim when disassembling an unknown lock.)