What is the NEC-compliant way to add a conduit in this configuration?
Type

ENT or "smurf tube" is described in NEC article 362 and seems right for this application. It can be bent by hand unlike EMT and does not require solvent-welding like PVC does.

Size

NM 12-2 is 0.400" in width. Chapter 9 Note 9 says to treat elliptical cables as single conductors with the diameter of the cable's major diameter. So each NM 12-2 is treated as a 0.126 in^2 conductor. Six of them equals 0.754 in^2.

Six NM cables counts as "over 2 wires," so we use 40% fill. For ENT, we consult Chapter 9 Table 4, which says that 40% fill in trade size 1-1/2 is 0.774 in^2, which will (just barely) accommodate six NM 12-2 cables. The next size up is trade size 2, for which 40% fill is 1.282 in^2. If you might pull seven cables, pick the larger size.

Termination

300.15 says "Where the wiring method is conduit...a box or conduit body shall be installed at each...pull point, unless otherwise permitted...." 300.15(C) says "A box or conduit body shall not be required where cables enter or exit from conduit...that is used to provide cable support or protection against physical damage. A fitting shall be provided on the end(s) of the conduit or tubing to protect the cable from abrasion. So the choices are box or bushing.