Any details about the captive/public wifi login screen browser? Is it just controlled by Safari's data. cookies, etc. somehow?  I'd think not since that'd be a major security hole. 

In any case, are there any tools for deleting the cookies from the browser used by the captive/public wifi login screen?
I use following to prevent Captive networks

Change: File Name

Captive network assistant


to

No More captive network assistant


in System/Library/CoreServices

Make note of this so you can change it back one day if you need to.

This will not delete cookies you already have but will stop getting more of them.