Kevin,
Wow, a 26-mile hike in for a caving trip! I did a fair bit of caving with the Diablo Grotto in California in the 1990s but never tried to do it UL (and even brought a hot tub to one state convention in Kings Canyon NP). At least some things are lighter now: helmets, LED lights, and lithium batteries all help.
The first thing that came to my mind, like Erik’s, is a pyramid like the Black Diamond Mega Light. It’s 4-person pyramid tent at 2#13oz, great in the wind (it’s our preferred shelter on our Aleutian Island trips), $290 with free shipping on Amazon. If you’ll be in a forested area and can reliably find a 57″ stick or tie off to the trees above, you could even cut out the weight of the pole. Then with 86″ x 86″ of floor space (bring some polycro sheeting), you’ll have lots of space to air out / dry out your gear. At under $300, it’s hard to get something truly top-notch that you won’t be tempted to replace later, but the Mega Light is, for some tasks, the best, lightest choice. If you’re later looking for a smaller, lighter solo tent, you can add one of thru-hiker choices like the Zpacks Duplex in the future when you have $600 laying around.
Spending 10 days in one spot would, for me, make it worth the extra pound to have 3x more room than the lightest possible option, give you space to dry out and sort gear, and some redundancy if someone else in the party has a tent failure.