Oh. So many options, so little time. LOL
So it sounds like you want to get the "high sierra" experience of the JMT in just a few days. Art's suggestion isn't bad.
With such a short time (and this depends on *your* personal foibles) I personally would not start/end from Yosemite Valley. If you do that you will spend most/all of your time on the part of the trail with both the lowest elevation and the largest number of people. You will not really make it to the true high country if you do JMT starting at Yosemite. ... On the other hand, like most of the trail, and in spite of the crowds and the lack of above-the-treeline scenery, the beginning of the trail would still be amazing. If this is your first trip to Yosemite I doubt very much if you are going to care too much. Your dropping jaw may distract you guys just a bit! No time to get jaded.
So decide first if you want to see a smidgen of the "real" high country, or if you and your group will be perfectly happy with the lower elevations and people on the beginning/end section of the JMT.
Personally I like to have my car at the end of a multi-day trip, and do the shuttle at the beginning rather than the other way around. So I would suggest one of these two based on which you think would be good based on what you said in the OP.
(1) (Art's) Stay night you arrive in Tuolumne Meadows camp, leave car at Tuolumne Meadows shuttle early next day to Red's. if timing is different then drop off car anx shuttle to red's first night. Long term parking is just to the east of bridge over Toulumne river on north side of Hy 120.
(2) Leave car in Yosemite Valley long term parking and take YARTS shuttle bus to Tuolumne.
Advantages: (1) less crowded, contain one "real" pass and some "real" high country. (2) Down hill, options to do Half dome (or better Clouds Rest), end in Yosemite Valley if you are planning to see that anyway.
Disadvantages: (1) No Yosemite Valley end (2) More people, *especially" last day, miss the "real" high country.
For 4 people, unless you are really pinching pennies, you will probably get the most time to backpack with the rental car. So entry/exit to and from from Yosemite from airport via Hy 120. Agree with Art on timing. June can present certain issues that are variable in the high country. August = no/few mosquitoes and no possibility of late snow on the higher route.
Edit: Think I may have misunderstood your time constraints first time. If you *are* free other months then for many September is considered the *best* time to go. After labor day the traffic will drop off hugely. In the high country it will be almost invisible. I did more or less the route Art suggested in late September and I think I saw one other solo and one couple the *whole* trip. So if that might be your thing then do it in September. Empty, mo bugs, good weather, but shorter days, and a bit colder.