embellished "indomitable hunter" entry for June 22, 1845, compared to plain accounts by fellow officers

To see how extraordinarily elaborate and literary the indomitable hunter passage really is, we probably should compare it to the straightforward reports for the same day by fellow officers and diarists J. Henry Carleton, William B. Franklin, and Henry S. Turner.

Carleton's is the closest to Cooke's.  Indeed, Carleton's narrative, originally published in the New York Spirit of the Times, looks like it might be the main source for Cook's expanded treatment.  Carleton is the only one (besides Cooke) to mention Captain Moore by name, when describing highlights of the previous day:

On the evening of the 21st, a large grizzly bear started up immediately in front of the command.  Captain Moore succeeded in striking it with a charge of buckshot, and chase was immediately given to it by several of the officers.  After an exciting race, the bear gained a small thicket, and finally escaped--though not without making three or four fierce charges from its cover at the horsemen, and scattering them right and left.  The buffalo range was again struck on the 22d., and the flesh of two fine cows was brought into camp by the hunters, after night-fall.
-- J. Henry Carleton, The Prairie Logbooks: Dragoon Campaigns to the Pawnee villages in 1844, and to the Rocky mountains in 1845, ed. Louis Pelzer (The Caxton Club, Chicago, 1943) page 256.

Yes, that's it!

Lieutenant Franklin also wrote about the memorable confrontation with a grizzly bear on the 21st, but gave no information about the buffalo hunt of the following day:

June 22.  To-day we were to leave the Platte and strike for the Sweetwater.  1 1/2 miles from camp we crossed the river and gradually left it.  In about 12 miles we reached the bitter spring where there was some good grass but very bad water.  It was a very hot day and this water almost as nauseous as sea water was delicious to us.  We stopped here a couple of hours, and then went to a spring 13 miles further, where we encamped.

For the first 12 miles the face of the country was comparatively smooth, that is there were no rocks.  But during the last 13 miles I noted limestone of various degrees of hardness, and now and then a granite boulder.  Some of the limestone has numerous impressions of shells in it.  The distance today was 25 miles, and the direction for the first 12 miles, 10 S of W, for the remainder of the distance nearly SW.

We noticed many salt efflorescences during the day's march, accounting well for the bitterness of the water. --William B. Franklin, March to the South Pass, ed. Frank N. Schubert (Office of the Chief of Engineers: Washington, D. C., 1979) page 17.

Like Carleton, Turner described the confrontation with the grizzly bear by the command.  Carleton and Turner agree that the command encountered the grizzly on the 21st of June.  On the next day, buffalo meat was furnished by the designated hunters (not the command that included Cooke's fellow officer, dragoon captain Benjamin D. Moore):
Camp N. 32. June 22. Crossed the river for the last time on the outward march—depth about 3 feet—width about 300 yards. Marched in a N. westerly direction, leaving high & bald ridges on our left; arrived at the “Bitter Spring” about 11 A.M. this spring takes its name from the peculiar saline taste of its water. Our animals however drunk of it freely. Our guide had promised us good grazing at this point, but on arriving at it we found it grazed to the roots by the Buffalo: a large herd moved off as we approached it; having halted an hour to permit our poor animals to feed as they best could, we continued on 13 miles & halted for the night at a spot which promised good & abundant water, but nothing else. The grass usually good is routinely eaten up by Buffalo. So far as our animals are concerned, & our well being just now is very intimately connected with theirs, we have indeed a gloomy prospect just now. Our grazing must improve, or they will scarcely be able to return to Laramie. We hope for much from the valley of the “Sweet-Water,” which we must reach tomorrow. Many Buffalo seen to day. The command has been subsisted for several days by the hunters. Distance 27 miles. Direction N. of W. Weather cool & clear.  "Journal of an Expedition" at fold3
(Letters Received by the Adjutant General, 1845 Kearny, SW, K113 page 52) Saw lots of buffalo, hunters come back with buffalo meat--and basically (again) that's it!

Three contemporary accounts of the same day's march, each in its way serious and relatively straightforward.  As you can see, none of the three participants described anything like the fierce pursuit by a "wild huntsman" that is so extravagantly fantasized in the August 1852 installment of Scenes Beyond the Western Border:

Amid the mirage and white dust, and the dizzy glow of reflected light and heat, which nearly turned the brain, I have still in my mind's eye a kind of vision of the indomitable hunter, Capt. M., scudding over far black slopes, which seemed themselves in wavy motion, fiercely pursuing flying buffaloes: it was a rivalry of all the German extravagance of their favorite legend of the wild huntsman. The facts seem simple, but there was an unnatural strangeness, a suffocating, alarming heat in the dazzling plains, and the black hills, that gave a dreamy confusion and doubt to realities. Did then, the strange mirage cheat the senses with apparitions of a desperate hunter, on that wonderful gray horse, pursuing black monsters, far, far and indistinctly into the glowing haze?

After all, we knew it was Ben. Moore,* or the devil! But it had always been said that he would follow a buffalo to the abode—left to that imagination which here seemed realized.