With that long of a trip you might have difficulty not getting into snow. Anything over a high pass might not be a good idea, if you’re not snow savvy.
That said, Rae Lakes might be your hot ticket. There were people heading up there in late April with snowshoes and spikes. Glen Pass will have a well trodden trail stomped up it. Poles and microspikes and a morning start and you shouldn’t have issues.
I hesitate to send anyone that I want to like me anywhere close to Yosemite… That’s Memorial Day weekend. True, starting on the 26th gives you an edge, you’ll have your pick of trailheads. BUT, it will be a madhouse and there will be holy f’n crowds all over every day hike out of the valley floor, so doing either of the rim trails is going to be the polar opposite of a wilderness experience. The backpackers will all be on the Buena Vista loop and over at Hetch Hetchy, sweating along with teetering packs and hoping that the creeks will be crossable.
Rae Lakes has an advantage in that Woods Creek and the Kings have bridges over them, and the other creek crossings should be manageable. Crossing at Junction Meadow to visit Lake Reflection might not be manageable but as desirable as that side trip can be, it’s probably not necessary. So that would be my vote for a Grand Sierra Experience.