I unknowingly cut my finger several days ago while wearing a light colored beaver felt hat and got a couple of spots of blood on the hat while handling it before I realized I had a cut. Without having water or a clean cloth, I made a "field" repair by applying saliva to the blood spots with a finger tip, then scratching the stain off with my fingernail. The saliva eventually dried, leaving a very light brown stain, probably from spreading some remaining blood. After the spots dried, I used a piece of 400 grit sandpaper to lightly scuff the spots, which almost completely disappeared.
A couple of observations:
The blood wasn't completely dry, so I had a better chance of getting it out.
Although saliva is primarily for dissolving dietary starches and fats at the beginning of the digestive process, it probably played a role in since blood plasma is a sugar/fat/protein compound.
I have used fine sandpaper to remove surface stains on hats with some success. If the stain is not too deep a light sanding will sometimes take it out. The key is not to sand too hard or for too long, or else you can create other problems. A couple of light scuffs is all I give it. The method also works better on lighter colored hats that are relatively clean. Otherwise you wind up scuffing off a layer of dirty felt and leaving a light spot.
I was instructed by a Hatter to do that on the Ivory hat I purchased for my Husband, but so far have not had to do that. Hat sponge and brush has kept it fairly super clean. I have here now maybe 10 hat sponges, rotate using them each when cleaning a hat, then soap and water the sponges and rinse and squeeze them out to dry faster. Hat sponges can pull a lot of light staining out of a hat real easy.