In my search I found a Major Arthur Caldwell. The following might be of interest:

"Caldwell, Lieut. Gen. , Sir James - Lillyman, G.C.B. of Beachlands, Isle of Wight, b. 22 Nov. 1770; m. 18 Feb. 1794, Mademoiselle Maliard of Dole in France, and relict of Richard Johnston, Esq, and by her had,

I. Arthur-James, who d. 11 June 1848.
I. Eliza-Marie, m. to Edward Sulivan, Esq. - 3rd son of Sir Richard Sulivan, Bart., by whom she has two sons and one daughter, the latter m. to John, eldest son of Sir Harcourt Lees, Bart.

Sir James-Lillyman Caldwell, a general in the Indian army, and chief engineer of the Madras presidency, is son of Major Arthur Caldwell, of the Bengal Engineers, and nephew of General Sir Alexander Caldwell, G.C.B."

To me it seems evident, unless someone can dispute this, that Arthur and Alexander are sons of William Caldwell.
If anyone knows of other sons, please advise. It seems like I am finding these sons, one-by-one.

Jerry Gagnon