While it is labelled as a DMR Kit, the string ID for the upgrade (bm_wpn_upg_ass_m4_b_beowulf) indicates that it is based on an AR-15-pattern heavy barrel and a caliber change to using .50 Beowulf rounds instead of the regular 5.56x45mm.
This is an odd decision on many grounds since the .50 Beowulf round is a relatively heavy type of cartridge and thus is not very suitable for use in a DMR setup. While offering substantially greater stopping power and armor penetration capability compared to the 5.56x45mm NATO rounds, the strengths of the .50 BW are offset by having lower kinetic energy overall and poorer range (short-to-medium) compared to standard ammunition loads.
The CAR-4 and AMR-16 gain a significant amount of recoil per shot upon installation of the DMR Kit. In reality, designated marksman rifles does not necessarily have increased recoil over their assault rifle counterparts. This is especially true in the cases of converted assault/combat rifles as the caliber remains the same, and the increased kinetic energy from the extended barrel is nominal at most as the barrel twist rate (which dictates recoil more accurately than barrel length) usually remains unchanged.
The barrel alone does not usually indicate what designation the weapon has, the DMR qualification usually encompasses the rifle's various accessories such as bi-pods, grips, scopes and stocks in addition to barrels. A more accurate name for the in-game mod would be DMR Barrel.
However, it can also be assumed that the DMR kit makes slight alterations to certain internal components as well (i.e. the gas system), which would explain the massive damage buff and sharp decrease in stability.
The Private Party achievement refers to the modification as the "DMR Kit AMR-16", suggesting that it was originally only intended to be equipped on the AMR-16 and not the CAR-4.