I've looked and googled for so long trying to find evidence of a church in or near Kokee, but I haven't had any luck. My suspicion is the CCC camp beside the Natural History Museum and Lodge or the cabins nearby may have housed a church in the past. There are also a couple of camps used by religious organizations that could potentially have a church as well.
Nu'alolo and Awa'awapuhi Trail are both in that general vicinity and while they both look like out and back trails, I think there might be a trail that connects them somewhere, but both end up back near the Lodge. There's also a New Spur/Waipo'o Falls trail that's an out and back, but farther south. The rest of the nearby trails wouldn't make sense to stash a bag as they are trails he'd take to get from camp to camp and would need his pack for.
It still really bugs me he didn't have a known reservation for the night of the 13th and it appears whatever happened, happened on the 13th/14th depending on accuracy of eyewitnesses. Without a camp setup, he wouldn't have had a place to leave his pack/gear while out hiking. I would so much love to know if that was an oversight on his part, he couldn't get a reservation somewhere, or if there are places to camp that do not need a reservation. Maybe hunting camps or similar.