Yeah, I know it's crazy, but Mt Whitney is the most visited mountain in the Sierra except Half Dome in Yosemite, which has dayhike quotas, too. People were booking hikes from the north and south and exiting via the Main Trail, so they set a limit on the numbers -- 25 exit permits a day.
When you booked the permit, you should have started with the "exiting Mt Whitney" option. That would have set you up to enter on the North Fork, and then get the second permits to exit. When you stop in to pick up your permit, you can ask if there are any no-show or walk-in Exit permits available. Ask about day before and day after your planned exit as well (those are usually ok with rangers). You might get lucky.
Permits are checked regularly, but not everyone is checked. With the permit you have, you are set to camp first night as you indicated, and then you can go up and come back down the Mountaineers Route. Actually 99% of people using the MR do just that. The descent really isn't that bad.
