Be sure you use a 30A double-pole breaker to adequately protect your wiring. Make sure you provide an equipment ground and use a NEMA 14-30 receptacle.
The box can be metal or plastic. Some form of cable clamp is always required, it's just that most plastic boxes have an integrated clamp (that finger-trap style door).
If using NM cable, The cable must be supported and secured by staples, cable ties, straps, hangers, or similar ļ¬ttings designed and installed so as not to damage the cable, within 300 mm (12 in.) of the outlet box. If the wire is being fished through a finished wall and the wire cannot be secured in such a manner, this rule does not apply.
Pure Opinion
As far as selecting an actual box, from experience, I recommend getting a 2-gang plastic box (or something equivalent in size if you want to use metal). The larger the capacity the better because a NEMA 14-30 receptacle with four #10 wires is not only bulky, but can be a very stiff pain to manipulate. They even make boxes that are "specially" designed for this purpose.
If you use a 2-gang box, you'll want to get a faceplate that is designed to match (faceplates for NEMA 14-30 receptacles are also manufactured for 1-gang boxes).
Your inspector is right and wrong at the same time!
First off, the existing install is a violation of NEC 422.16(A), so your inspector was correct to point that out:
(A) General. Flexible cord shall be permitted (1) for the
connection of appliances to facilitate their frequent interchange or to prevent the transmission of noise or vibration or
(2) to facilitate the removal or disconnection of appliances that
are fastened in place, where the fastening means and mechanical connections are specifically designed to permit ready removal for maintenance or repair and the appliance is intended
or identified for flexible cord connection.
As to the proposed fix, however, your inspector is confused or working from an old Code edition, as the 2014 (and 2017 AFAIK) NEC do not require a disconnecting means for a hardwired appliance if the breaker can be locked off (an electrical supply house will be able to order in lockoff devices for any modern breaker type). This is 422.31(B) in the Code:
(B) Appliances Rated over 300 Volt-Amperes. For permanently connected appliances rated over 300 volt-amperes,
the branch-circuit switch or circuit breaker shall be permitted
to serve as the disconnecting means where the switch or circuit breaker is within sight from the appliance or is lockable in
accordance with 110.25.
So, I'd simply take an ordinary single gang metal faceplate with a 1/2" KO in it and use that instead of faffing about with a box extension that needs grounding. (250.148 calls out metallic boxes, but not metallic faceplates on nonmetallic boxes...although it is still wise to attach a grounding pigtail to the faceplate in this case.)
Best Answer
This should be fine, but triple-check the fill if you want to be sure, and make sure it's labeled properly too!
Your usecase (extra wires run through a disconnect box to another disconnect) falls under NEC 312.8:
The 40% and 75% fill rules are trivial in this case -- the gutters on even the smallest disconnects are larger than 3/4" EMT in order to provide the Code-required wire-bending space, so you'll be nowhere close to either one. Since these are true feed-through conductors, though, the existing disconnect will need a a label applied to it in accordance with 110.21(B), though:
Given all that, I would simply make up a printed label that says "Caution: Feed-through conductors are disconnected by breaker XYZ in the panel" where XYZ is the label given to the new breaker in the loadcenter's directory, and apply it to the inside of the existing disconnect's cover where it's sheltered from UV, water, etal.