>> taking each clump out of the bed and placing (perhaps on a large tarp) in the shade.
...
>> Then move the lilies back into the nice rich clean bed
This reminded me of something I read in a garden book from the 1940s or 1950s. I Googled "heeling in plants" and I was pretty close. I think classic "heeling in" is done to bare-root trees or bushes to hold them for a winter or a summer, when a permanent site is not available.)
If you need a few days to dig up that whole bed, screen out grass roots, and rejuvenate it, the daylilies might appreciate it if the roots were somewhat protected by heeling in (with or without a tarp, but surely with shade).
The tarp were laid down over a shallow trench and the excavated soil, then the daylilies laid down at an angle, tops resting on the tarp over the excavated soil and roots on the tarp in the trench. Then the roots could be loosely covered with some light and easily wetted mix and kept damp.
But it would only be worth digging the trench if re-doing the whole bed were going to take a few days.
(Or if you needed to uproot all the grass where that trench was, anyway. Or if you wanted to widen the bed by the width of that trench.)
