I believe that you want to select "Overnight NF visiting Whitney." Even though you will not be taking the NF to Mount Whitney, you will be entering the "Whitney Zone" past LBS Lake.
As for the other options, "Overnight NF exiting whitney trail" is for people who stay overnight on NF, summit, and return via the main Whitney trail. Those permits are subject to the Trail Crest Exit Quota in addition to the NF entry quota.
Cross country implies no entrance on the NF.
