I love the Whites but this is a trek and half when your plan is do Mt. Whitney that weekend. I'd opt for Horseshoe Meadow.
I would, also, not worry about the bear box storage in Horseshoe. There are boxes all over the places...in the camping areas and at the trailheads.
My favorite trip to Trail Peak would be the Cottonwood Pass Loop. Head to CP then go left. Trail Peak is a relatively easy climb from the trail then angle off it to the Trail Pass. Then back to the trailhead.
