Hi all! Thanks for the input, unfortunately I still haven't been able to fix the issue. I'll give specifics under the below quotes:
It looks like there were still some old registry entries from CKAN. I ran CCleaner and I'm not seeing CKAN in dxdiag.txt anymore, so that doesn't seem to be the issue.
Updated dxdiag.txt: https://www.dropbox.com/s/ah87izb2paib544/DxDiag.txt?dl=0
KSP.log: https://www.dropbox.com/s/kvyl8ilnunac4d6/KSP.log?dl=0
So I was pretty sure my drivers were up to date, but sure enough Intel had a newer driver on their site that I wasn't getting through Dell. I applied the update, but it's still crashing.
Updated Player.log: https://www.dropbox.com/s/7uxvlx9d1v8k0h4/Player.log?dl=0
I uninstalled and reinstalled KSP again after cleaning up CKAN, and it doesn't seem to be generating error.log anymore.