I've noticed a bad vibration that basically shakes the whole car as I release the clutch in 1st and give it gas. I'm pretty sure it's coming from the rear end, but the whole car shakes and vibrates and it only happens in 1st gear from a stop. It feels like I'm driving over a series of road reflectors, then it goes away once the clutch is fully engaged.

Any thoughts?

Your flywheel is warped from excessive heat. The clutch will grab a high spot and then let go again. My daily driver does this. Coincidentally it started right after teaching my wife to drive a stick.

Thanks Mike, that makes sense. I believe I first noticed it after I ran the dyno at the MSA Nationals, where I was pushing it pretty hard.