If you want to spend a couple nights up there for acclimatization - you may want to reserve a 'regular' site in the main campground. Book early though, as they fill up pretty quickly in my experience
As Tif wrote, the campground often fills up way before you get an answer from the lottery. I hear Inyo is switching to a modern online permit system, so maybe someday they can coordinate these two reservation timelines. I would plan on spending the evening before your hike at the portal somewhere with our without a camp reservation.
If you have more time to acclimate, I recommend spending a day or more at the Cottonwood Lakes trailhead parking area (10,000 ft). Nice warm-up hike to Cottonwood Pass (11,000 ft) or beyond. They have hike-in spots with bear boxes. I like to acclimated there, but I sleep better at lower elevation at the Portal (8,000+ ft). I have a camper shell truck so its easy for me to move around, but there should be room for a tent somewhere at the Portal.
Just be flexible and creative and it will all work out. Good luck.
