I would add gloves, and make sure they are the heavy-duty nitrile variety because they are sunlight, oil (sunscreen and bug juice) resistant. IMO three pair is the absolute minimum. More is better. Six pairs is what I shoot for.

Also tape, roll gauze, and gauze pads go very quickly. Tape gets used patch and hold. Half a roll of high quality duct tape us a life saver. Medically or otherwise. The pads get used for everything from basic cleaning and padding to actual use as bandages. Roll gauze gets use by the yard to wrap, hold and pad. Hard to carry too much of any of those.

I would skip the backup battery for the thermometer. Put in a new lithium cell before every trip. One less thing to rattle around. I would add a key-chain flashlight to the kit and the smallest EMT shears you can find. Light and shears get dummy corded to the kit.

I would add at least one triangular bandage. For quick and dirty, on-the-fly bandaging the all-in-one ease of use of the triangular bandage is a time saver. It also works as a tourniquet, large burn cover, and a work sheet to work off of. Quickly tied on to keep the blood in and dirt out you can come back and do a hospital-grade clean and bandage job later.

I would read, and reread, the book until I knew the material. I would take simple notes and print them out of waterproof paper or laminate them. The notes should outline sequences and be a reminder. Tuck the cheat-sheet into the kit and leave the book behind. Use the space and weight saved to stock up on gloves, duct tape, gauze, and triangular bandages.