Early May is a good time in the Grand Canyon. Too late for Death Valley. Too early for the White Mountains or Sierra.
R2R is a great hike, but, yeah, requires some substantial logistics to get back to your car.
Personally, I'd rather be hiking than driving the same roads twice and spending a fair bit on shuttles or nervous about hitch-hiking. In round numbers, Vegas-SR, Vegas-NR and SR-NR all take about 4:44 to drive.
If you've done the Zion Traverse, you have a very good sense of how you hold up over the miles, the vertical climbs, descents, and how much conditioning you need in advance.
I'd consider, in increasing levels of difficulty:
South Rim (SR) - River - SR: lots of easy lodging (I usually use Red Feather Lodge in Tusayan just outside the park if I want a real bed cheap or last minute), easy to get supplies at markets on the SR, expansive views especially on the South Kaibab trail, 1000 feet less vertical than from the HR, albeit with many more tourists along the way. Start early. Like 5 am. You can't start too early - 4 am with a headlamp isn't wrong. Wear a layer less than you think you need because 1 mile in, you'd peel it anyway. Starting early avoids all the tourists on the way down and much of the way up. And you get up and out before the heat of the day hits. Start early.
NR-River-NR: Higher, cooler, far less hiking traffic. Going to the NR lets you access Zion easily to fill out your 4 days with day hikes.
SR-River-a ways up along Bright Angel Creek-SR - I did that two years ago on a BPL R2R2R because I wasn't up for the whole enchilada (hard to train in an Alaskan winter). That could be 20, 25, 30 miles - however much you felt like in a day. Just turn around when you're 1/3 fried.
SR-NR-SR - I did that when I was half my age and it definitely is in the "death march" category. But old farts do it, too.
Everything I mentioned above is in the context of a "day"hike. Maybe a 20-hour day, but no permits required. Camping permits are hard to get for the main corridor so you'd probably have to wait for the last-minute permits. There's a whole routine of waiting one day for a low number in line the next morning but it sounds like you don't have a really long time.
Something I've done from the NR is a "hike till you're tired, sleep till you're cold, wash, rinse, repeat" thing with a friend. Mostly we did it to fit Half Dome in Yosemite, Angel's Landing and Narrows in Zion, and hike to River in GCNP - all in 3 days. Bring 10,000 calories, extra batteries for night hiking and a sleeping pad. DON'T look like a backpacker who needs a permit. Never admit to sleeping or being ready to. Just get well off the trail and nap a little. In tan clothing.