You give me the opportunity to answer a very delicate question.
Every technician gives a certain value to their own work. Everyone has trained over the years, investing time and money, and has acquired a "certain" expertise. Not all technicians are often prepared for every type of intervention. Some know how to tune very well, but do not have experience, for example, with voicing. Some think about "standardizing" regulation and/or think that what their father or grandfather said was the truth. Some pursue personal philosophies that distinguish them and would like to perform a certain intervention on all the pianos they visit, because they think that everything or much depends on doing or not doing this or that.
I believe that everyone should intervene according to their own experience, trying to understand, first of all, what the pianist requires most urgently. It is, however, professional to analyze and "examine" the entire instrument, from top to bottom, and reveal everything that would be necessary "to do".
In this scenario, it is not easy to say how much a "tuning" might cost. Often the Tuner or, better yet, the Technician, is called for a "tuning," but many times they find themselves having to resolve other problems.
Every intervention must contend with competence, time, materials and equipment used, and the type of piano.
It is difficult to establish from a distance even the cost of a single tuning.
Sometimes a piano has not been tuned for a long time and has drifted from pitch. Or it has very old and disharmonious strings. Or.......
I know you will shrug your shoulders at this response of mine, but believe me, the Technician's honesty and professionalism must dictate the impossibility of standardizing an intervention cost. Naturally, the distance to be traveled to reach the client also has an influence.
I also believe that some time should be spent talking and deciding together how to articulate the good maintenance of the instrument with minimum expense and consolidate a good mutual relationship of trust.
P.S. The issue with the partial in question is certainly highlighted by your sensitivity to that partial, but if it also translated into harshness of the fundamental note, then it would be well-intoned. It could also depend on an imperfect alignment of that hammer zone according to the optimal striking point. All to be seen.... It could also depend on the release of the damper (too late and not simultaneously across its entire length, for example... check it!!)