I was considering the same variables you are just a few days ago. I had done the regular trail 4 times, but was going on the MR for the first time starting on the 28th of July. I never had altitude issues on the regular trail but I know the MR is twice as steep, so I wanted to be more cautious. I ended up driving from San Diego on the 27th and spending the night at Whitney Portal. I chose to hike to UBSL on the 28th. I could have easily gone to Iceberg, but I was concerned about the night I was going to have up there. On the 29th, with a light pack, we sumitted and got back to the trailhead (started at 3:30 am and were back by 3:30 pm). It worked out well. By the time I started having a tiny headache from the altitude, I was already back at 11,000 ft or so. It disappeared by the time I crossed back at the 10,000 ft level. Because I spent the night at the trailhead, I think going to Iceberg would be OK. I would not try to spend the night at Iceberg if I just drove from sea level that day though. I think if I were you and the schedule was tight, I would spend the first night at LBSL at 10,000 ft and summit the next day, getting back to LBLS as quickly as you can.
Best of luck. The MR is a gorgeous trail!