Have you mentioned your worries to your neighbour?We've decided to shorten our weekend to just Friday morning to Saturday evening. So we need one less day of chicken tending from the neighbor. Still, this coming week and then who knows how long, is worrisome.
We don't have a shade cloth. We have been hosing the building down every hour or two. Most of the day it is in the shade, but in the afternoon it does get a lot of sun. The only days we won't be able to do that is that Friday/Saturday.
We have one box fan and one smaller battery fan. I'll check and see if any neighbors have more. I may try to take down some of the walls myself and replace them with chicken wire. I know that would help air flow lower down.
I can't put a hose on a timer. For lots of reasons, my well system won't allow that. I'll have to hope our neighbor will take care of them.
We won't have this problem next summer. The new coop will fix that. But it's not ready now. It's not predator proof in any way...its just a shed. It can't be ready any time soon.
I'm wondering... worst case scenario... what if I tell the neighbor to just let them out of the run? Will they go find a cooler place? More importantly, will they come back? I don't know where that leaves Eenie and the triplets. They'll be 2 weeks old on Thursday.
Most people are more than willing to go the extra mile I have found. I bet if you explained to the guy which hen is extremely sensitive to heat, and the babies need a fan on them - I bet he would be more than happy to help out.
My horse neighbour and I regularly help each other out with our horses. Both of us have been known to just jump in our trucks and pop over when we have emergencies.
Te shade: Even a tarpaulin used as a shade clothe would help. Or an old sheet, I have a neighbour here that placed an old sheet in front of his coop/run for shade, and it doubled as a movie screen in the summer

I would not let them roam due to predators, and if they are not used to being out that would be worse than maybe the heat.
Is there a Walmart nearby or Harbour freight? I got some nice box fans for my mums RV there when she used to go to AZ. Maybe your neighbours can loan you a few.
Worse case, I would throw an old sheet up over / in front of the run for shade. And put that fan on the babies. Are they running with the flock yet?
Shade: https://www.pinterest.ca/pin/victorian-greenhouse-vi46-in-2023--384213411975337804/