Wilderness Permit Pickup in Yosemite Valley

I am leading a group that will be driving up to Yosemite from the Bay Area middle of next week and backpacking out of the Happy Isles -> LYV trailhead.

Our typical routine is to use the Big Oak Flat entrance station as a rally point where all the vehicles meet up, we pick up our wilderness permit there, and then caravan to whatever trailhead we are departing from. With Hwy120/BOFR closed inside the park, we'll have to travel via Merced and Mariposa this time. I plan to make El Portal the rally spot where we meet up and caravan from there. The problem I'm trying to work out is how best to proceed from that point.

The best I can think of is to have all the vehicles enter the park and drive into the valley along Southside Drive, across Stoneman Bridge, and back to the main Day-use Parking area. I run down to the Wilderness Center to pick up the permits while the rest of the group looks around the store/village for a few minutes. Once we are back at the cars, we drive west along Northside Drive. Sentinel Bridge is supposed to be closed for construction, so we'll need to go all the way to El Capitan bridge, cross over the Merced, then work back east along Southside Drive to Trailhead Parking where we can finally load out and start hiking.

Can anyone think of a quicker alternative to that method? I'm just worried it will be after noon before we start hiking. At least we have a relatively short hike to LYV. And at least we won't be facing weekend traffic!

The only alternative I can think of is to send one person up a day ahead of time to pick up the wilderness permit and stay overnight in the backpackers camp, which would allow the others to drive straight in to Trailhead Parking the next morning when they arrive. I'm considering that, but not sure we'll be able to make it work.