Your dog is affected by a phenomenon called “post-clipping alopecia,” which (as you acknowledge) can occur after a dog is improperly groomed. (All emphases are mine.)
This is a condition that is very common in the fur bearing breeds. The fur bearers such as Alaskan Malamutes, Keeshonds, Siberian Huskies, Samoyeds, Pomeranians, Chows, Labradors, Golden Retrievers,Shepherds,Shetland Sheepdogs, Newfoundlands etc. all have undercoat that needs to be brushed on a regular basis to keep shedding to a minimum and to keep the dog from getting matted. Post Clipping Alopecia comes into play when a fur bearer is shaved down. The hair will NOT grow back to the way it was. It can come close, but it will never be as beautiful as the original coat.
The rest of the above source seems relatively non-scientific and includes no citations, so I’ll move on. In terms of treatment, some academic literature encourages a hands-off approach:
Symmetrical alopecia in the dog, Rosario Cerundolo, Companion Animal Practice, 1999, 21,350-359.
POST-CLIPPING ALOPECIA does not require any therapy as the hair will grow back after a few weeks or months.
Based on your description, this does not appear to be true for all cases of PCA, and other, more recent sources refer to active treatments:
Post-clipping alopecia is the failure to regrow hair for months after clipping, usually after surgery or grooming... The area over the back and rump is most commonly affected, and has a "just clipped" appearance, even months later. The reason the hair fails to regrow is unknown; it may be due to vasoconstriction of blood vessels in the skin that occurs with decreased skin surface temperature from loss of haircoat insulation after clipping. Histopathology shows a predominance of catagen hair follicles. Total hair regrowth may take six to 24 months, although most animals regrow hair within 12 months. Initially the regrown hair may be darker than normal.
Differential diagnoses that should be ruled out with appropriate laboratory testing include causes of endocrine alopecia such as hyperadrenocorticism, hypothyroidism, and sex hormone imbalance. Vigorous brushing of the area, massage, hydrotherapy with warm water, and covering the alopecic area with a sweater to increase skin temperature and blood flow to the skin, may stimulate hair regrowth in some cases. Melatonin 3-6 mg orally every eight to 12 hours for eight to 12 weeks and short-term treatment with levothyroxine at 0.02mg/kg orally every 12 hours for four to six weeks may also increase hair regrowth in select cases.
From what I’ve read, the treatments listed above are essentially the extent of your options. However, other hair-growth treatments that are effective in humans might be viable therapy candidates (under veterinary supervision in a controlled environment) if your dog fails all the options listed above.
As mentioned, a working hypothesis for the pathogenesis of PCA is vasoconstriction in response to reduced skin temperatures (from hair loss); Minoxidil, for example, is a well-characterized hair-growth agent used in human topicals because it causes vasodilation (which is the goal of many of the non-pharmacological approaches to treating PCA like massage or vigorous brushing), but it toxic to dogs when ingested so it would be a challenge to deliver in a safe manner.
That said, you might also consider your values relating to your pet. Does you dog’s PCA bother you to the extent that you are willing to compromise its quality of life to meet your aesthetic standards? Most of the treatments I’ve mentioned are completely harmless and your dog won’t mind them, but if they don’t work, I’d encourage you to reconsider attempting to improve your dog’s coat and instead focus on its happiness.