... If we pack up and leave Yosemite by 10-ish, we'll get to Lone Pine around 2 so it'll be well past the 10 am same day overnight permit pick up. Unless you can still obtain one later?
Yes, you can obtain one (assuming one is available) anytime AFTER 10. More are released at 2, so between the two release points during the day, you stand a reasonably good chance.
The more snow accumulates, the better your chance of getting a permit. People tend to bail when there's whitestuff underfoot.