Interesting that your books say that they wanted a lighter gun. The following comes from the procurement order for six guns dated 8 Oct, 1863:
"...three pairs of guns to be made in strict accordance with the following directions: each pair to be made of the same metal at the same time and place, under precisely the same circumstances. One of the guns of each pair to be a 30-pounder banded gun to fit the present 4.5-inch siege carriage; the other gun of each pair to be similar in outline to the 4.5-inch gun but also bored to 4.2 inches, and weight of each no to exceed 3,550 pounds. One pair of the guns to be rifled with uniform twist; the other with a gaining twist like the present 30-pounder... The third pair of guns will not be bored, but be left unfinished subject to such orders as may be given on the subject hereafter." The first two pairs of guns were delivered, along with one of the last pair.
There are documented weights for three guns from this order: foundry no. 405 (3,590 lbs), 407 (3,610 lbs) both banded, and 806 (3,560 lbs.) The data lets us assume the Eaton Rapids gun is foundry number 407, inspected by DWF on 1 March, 1864, delivered on 16 July, 1864.
The other two guns delivered where "similar in outline to the 4.5-inch gun", foundry numbers 416 & 421. Both were inspected by RMH on 23 July, 1864, but not delivered until after the end of the Civil War on 6 September, 1865. You can't help but wonder if they are out there masquerading as a 4.5in Ordnance rifle.
The 4.5-inch Ordnance rifle weighs about 3,550 pounds, the 4.2 inch Army Parrott weighs in at about 4,200 pounds. The Navy version weighs in at around 3,500 pounds, so they already had a "lighter" version of the Parrott. Was this a case of trying out different rifling or perhaps providing technology to West Point to cast better cannon?