I could always carry a water bottle to replenish the very small amount of water that I can manage without too much sloshing but I'd rather see how much water can be conserved with a lid and try to forgo carrying a water "tanker" with me. By the time the water level is low enough with only one stone in the holder (providing more water than I'd have with additional volume of a second stone), I run out of water after about 5 honing sessions with the Arctic Fox just due to wetting the stone enough to keep it clean in very hot weather. Thus, even with your schedule, I'd probably run out of water about the same time I'd need that Bull Thistle. To date, I've been filling to a higher level to accommodate more honing without a holder refill and I just live with slightly wet pants and t-shirt. Like I said, it's been hot. As you can see from the photos, our terrain is somewhat hilly. Perhaps marching up and down the hills or chasing down a distant wild rose bush causes me to use more water, even though I'm trying to walk and move so as not to slosh (which somewhat hampers my style).
I keep my stones in a water bottle with a cut-off top so they never soak up any of the day's initial dose of water. In order to carry two stones, I think a one-stone lid usage model might be to carry the Bull Thistle dry / damp in a pocket and only wet it when needed. I don't think the smaller inside overhang achievable with a two stone lid will be large enough to reduce sloshing much (the two-stone rectangular hole will be too large), but I might be wrong about that so I'll make both version of the lid.
I have a bunch of the old Taulman 618 Nylon 3D printer filament and it has just the right amount of flex for a snap-on lid. Nylon is also very rugged and won't get soft or even gooey in the sun like PLA. ABS or PET-G have sufficiently high glass transition temperatures too and they would probably have enough flex to snap onto the holder too. This is actually a trivial 3D printing project and I like to design and 3D print useful things, rather than downloading garden gnome trinket models and such, which is mostly what people do with their 3D printers.
I'll post photos here once I've printed them and I plan to send both single and dual stone versions to you, if they both prove to have any merit.