Hi Gus:
I'm not certain what to make of the "war eagle" theory. Unfortunately, army regulations are rather vague on this subject. The following is the 1897 regulation describing the undress cap.
10. Of dark-blue cloth, the diameter at the top slightly less than at the base, the height 3-1/4 inches all around; the seam around the top without a welt, and neatly stitched on each side. The band 1-1/2 inches wide, with welts projecting 1/8 inch at the top and bottom, the bottom welt 1/8 inch above the base of the cap. The visor of black patent leather, bound with the same, and molded to shape, green underneath, rounded and sloping downward from the horizontal. A rigid stiffening all around to extend from the base of the cap to within 1 inch of the top, and a hair-cloth stiffening throughout the remainder of the sides and top; four black metal eyelets for ventilation; two on each side, placed above the band; a cap cord of gold bullion 1/8 inch in diameter, secured at both ends by small regulation buttons, one on each side, immediately back of the ends of the visor. For general officers, a band of black velvet, and for all other officers, except chaplains, a band of lustrous black mohair braid filling the space between the welts.
The cap badge for all officers will be the coat of arms of the United States, embroidered in gold, modified according to pattern, and will be placed in front so that the top of the badge will be slightly below the top of the cap.
I believe Stuart's Chaplain cap predates 1895 because of the bullion cross device. In 1895, branch of service insignia would have moved from the cap to the standing collar of the sack coat and been replaced by the coat of arms.
Chas.