I don't believe they are omnivores but stoats and ferrets are also a huge pest in New Zealand.
DOC controls them in two ways - firstly by controlling rodent numbers, secondly by directly controlling mustelids.
The rodents - a spike in mice/rat populations causes a spike in mustelid populations. Then when they run out of rodents to eat in the forest, they go for the birds. And our native trees (the Nothofagus beech trees) often "mast" - that is, all the trees drop their seed all at once, which is a major food source for rodents. So a mast year will always lead to a massive spike in mice and rats, which then leads to a massive spike in stoats and weasels.
As for mustelid control - it's primarily done via trapping (although there can be significant mustelid bykill from large poisoning operations targeting rodents or possums). DOC has a standard trap that works well when baited with an egg - but trapping is very manpower intensive and DOC is not very well funded.
* Deer (red, fallow, sika, rusa, whitetail, and wapiti/elk) - largely managed by recreational hunting and/or "wild animal recovery operations" (WARO) - where commercial hunters shoot them from helicopters, retrieve and process the carcasses and then ship the meat to countries like Germany that love a bit of wild venison
* Chamois - managed primarily by recreational hunting, with the occasional WARO to supply the market for "gamsbart" in Austria
* Tahr - managed by recreational hunting, especially trophy hunting (one of NZ's premier trophy species), but also culled by DOC when numbers get too high for their environment
* Wild pigs - mainly recreational hunting, some control by AHBs to prevent TB.
* Wild goats - recreational hunting and culling by contractors working for DOC
* Wallabies - Unsure of any official eradication plans, but often hunted recreationally
* Wild horses - Only really exist in the Kaimanawa ranges. Population managed by DOC. When overpopulated, they round up horses and resettle them with interested owners. If not enough horses can be resettled, they cull.
* Wild cattle - Mainly controlled by recreational hunting. Apparently wild beef is lean and tasty as hell. Very limited range, but I'd love to try it sometime.
* Rabbits - Recreational hunting, professional cullers, poison drops, biological controls (calicivirus etc.), and oh, that time that settlers released ferrets, stoats, and weasels into NZ to control the rabbits. The stoats and weasels decided that rabbits were way harder to catch than our native birds, but ferrets and feral cats still eat a lot of rabbit.
Not sure if I'm missing any other imported herbivores that could be considered a pest.