Here is a similar construct (with very similar meaning) where you can use the subjunctive I:
Der Verkäufer leistet dem geizigen Kaufmann einen treuen Dienst mit dem Wunsch, dass er seine Tochter heirate.
What is going on here is this:
The subjunctive is not triggered by wishes as such (though many grammar books falsely claim so or at least accidentally imply this), but by an indirect speech, or a very similar dependency.
A good litmus test for this is if you can phrase the same thing in direct speech, without changing the rest of the sentence:
In the above example, this works:
Der Verkäufer leistet dem geizigen Kaufmann einen treuen Dienst mit dem Wunsch: »Heirate meine Tochter.«
By contrast, this doesn’t work with your sentence:
* Der Verkäufer leistet dem geizigen Kaufmann einen treuen Dienst (damit): »Heirate meine Tochter.«
Here is a similar construct where you can use the subjunctive II:
Der Verkäufer leistete dem geizigen Kaufmann einen treuen Dienst in der Hoffnung, dass er seine Tochter heiraten würde.
Here, using the subjunctive II implies that the merchant did not actually marry the daughter. It is relevant that everything happens in the past so we can now what actually happens. Another requisite for the subjunctive is using in der Hoffnung instead of damit. The former leaves room for the possibility that the marriage does not happen. By contrast, damit implies a direct cause-and-effect relationship to some extent and therefore cannot trigger a subjunctive.
Finally note that a similar, archaic conjunction that triggers the subjunctive is auf dass:
Der Verkäufer leistet dem geizigen Kaufmann einen treuen Dienst, auf dass er seine Tochter heirate.
This is a fossilised construct and therefore it is best to learn it as such instead of trying to explain it by modern rules.