I rode the Cloud Cap road in Oct 2010 on my steel road bike with 28mm tires and 36 spoke wheels. We were on a four day tour from Portland and rode from Hood River to the Tilly Jane cabin where we spent the night. I went with a friend who rode a Surly Steamroller with 32mm tires and jury-rigged 1x5 gearing. My friend was admittedly a little crazy to go up there on that bike. He did fine though.
With low enough gearing- I was at 34-27 and wanted lower at times- it was fine, if slow going with a light touring load, including sleeping bag, clothes, food for three days, spares/tools etc.
All a mixture of loose gravel, hardpack, rocks, ruts etc. We were also lucky in that the road had a few days to dry out before we went up there. I could imagine it being very different (in a bad way) during or after some rain.
25mm tires might be not very fun, especially on the descent, but if you are experienced on rough loose gravel roads and willing to take it very very slow I would say it is possible. It would be worth it to go larger tires if your frame could fit them.
This is all qualified by the fact that I haven't been up there since 2010 so not sure if the conditions have changed since then- which is possible since there is a large burn area there that would presumably accelerate erosion of the road.