The clause "assuming all of the above is permitted by statute" introduces an invalid assumption. "Permitted by the statute" is (a) a question of legal interpretation and (b) only a part of the "due process" question.
In US law, "due process" means "follows the law". One source of law is the constitution (state or US). A second is a statute. A third is regulations that are statutorily authorized. A fourth is case law. If you don't follow the law, you have a due process problem. The government has to follow fair legal procedures in depriving a person of life, liberty, or property. If it doesn't, you have a procedural due process problem, and that could figure into a lawsuit (then the courts will decide if your claim is correct). Here is a bit of a guide.
The facts as stated to not show that the required fundamental fairness was omitted or that the law was not followed. If the law obligates the board to accept the judgment of an external reviewer, then there has been a violation of due process – there is no chance that there is any such requirement. If the board is legally required to state a reason for denying an appeal and they fail to do so, then they have not followed the law.
There is also "substantive due process", which involves fundamental rights. The right to ply your trade is a fundamental right "implicit in the concept of ordered liberty", but the Commerce Clause (in your favorite state) also gives the government the right to regulate businesses, so there is a balancing act that has to be performed. In this case, a board cannot just say "we don't want this guy working in our state because we don't like his face", so they would have to have a reason to deny the person a license. It might take a lawsuit to get them to reveal that reason, and the lawsuit would resolve the question of whether the government followed the law.
This article touches on Due Process concerns w.r.t. occupational licensing in Texas, and Patel v. Texas is a Texas case where the court found that a requirement imposed on persons to receive a license to pursue their trade was unconstitutional on the state equivalent of the Due Process Clause (a substantive due process issue). Here is a page on procedural due process, with various court citations, such as Mackey v. Montrym, 443 U.S. 1 where the fundamental issue was failure to have a process (a license was suspended without a hearing). Such examples might give you an idea whether in the particular case, there is a due process concern, in case this is not entirely hypothetical.