This question sounds to me like there is an inferred intent of the person giving the invoice to manipulate their value added tax rate due to the misclassification, and then an unrelated issue related to an overly high charge for the services which are basically independent.
Is the classification of the job material in some way to either of those issues in a non-obvious way that you haven't clarified in your question?
I will answer on the assumption that it is not, because at first glance, it is hard to see why this would matter with respect to either issue.
A CVR listing is sort of half way between a U.S. Secretary of State company listing and a U.S. Securities and Exchange Commission company listing with more than just bare bones contact information but less than a full fledged public company's disclosure statement.
The fact that the company's business type classification is not a precise match for the kind of work that they did for you on this job is not obviously material to either what VAT taxes would be due (which usually depends upon the particular kinds of goods and services involved in the transaction) or the rate that a company can charge for a particular kind of work done. The fact that a company is classified in one category that describes the overall enterprise doesn't mean that every last thing done by the company must precisely fit that description.
The mismatch could also conceivably be nothing more than a clerical error made by somebody preparing the CVR listing, in which case, again, the answer is "so what?"
Therefore, as far as I can tell from the limited information in the question, this is just an immaterial error with no relevance to any of the likely disputes that you might have in this case.
Usually, an immaterial error in a public record or invoice would not be a ground to object the amount that a company charged for the work, nor would it be something that would be appropriate to complain about in any way that would bring you an advantage that I can see in this transaction.
This doesn't mean that you don't potentially have grounds to object to the invoice and dispute the higher than expected charges. But, bringing up the issue of the CVR code doesn't appear to add anything to your rights in this dispute.
This said, I can certainly imagine circumstance where the wrong CVR code could be a hint that something else really is amiss.
For example, suppose that you need a license to do a cleaning job since it involves environmental hazards and waste disposal, but you don't need a license to do management consulting, because who cares if someone is stupid enough to take your bad management advice. In that situation, the company might be misclassified because it doesn't have and perhaps cannot get for some reason, the business license that it needs to do the cleaning job.
Charging for services while not having the proper business license might very well be illegal and a ground not to pay, and the error in the CVR filing might actually be an attempt to circumvent this problem. But, that kind of possibility relies upon pure speculation and without more facts there is no good reason to assume that something nefarious is going on here.