No, it would not be appropriate for Judge Harris to write the letter of recommendation because the judge may not use the prestige of the judicial office to advance Miles’ private interest.
(CJC Canon 2(B))

It does not matter if the letter is written on Judge Harris’ personal stationery because there is the potential that the letter could be used to lend the prestige of the judicial office to advance Miles’ private interest before the licensing agency.

In making the determination as to whether to write a letter of recommendation for Miles, Judge Harris needs to consider the reason and audience for the letter of recommendation.

In this case it is problematic because the letter to the state regulatory agency would be used to advance Miles’ private interest.