Yes, definitely some reinforcement on the inside, wood or metal strap. I think one of the factors here is the depth of the cabinets. Eighteen inches is pretty-deep for a wall-mount. and you should upgrade the inside strapping, in my opinion. Some 1/8" x 1" steel strap should do well, maybe thinner would be ok, but 1/8" is pretty common and is easy to find.
I hung seven and a half feet of 30" x 30" x 12" welded steel boxes from Whirlpool's Gladiator line in my garage. They ship w/a French cleat-style pvc strip to mount to the wall and a steel piece to mount to the back of the cabinet, top and bottom. Snce I was mounting the cabinets to a CBS/CMU wall, I just used a ledger cleat running the length of the bottom of the cabinets, on which the cabinets rested. Then I used 1/4" x 3" tapcons w/fender washers into the masonry wall. It was very solid, and capable of the light load I was putting into the cabinets.
Sounds like you're gonna get it done correctly w/the idea you had.