There seems to be a bit of a problem with the question. You have stated the properties of the random number generator, for example, that it is unbiased. However, if we were to make an objective interpretation, using an objectivist interpretation, I don't think we can say that the generator is unbiased if the experiment (i.e. seeing what number it produces) has never been done before, thus the only sensible interpretation would be a subjectivist one.
Note that once the generator has output the 7, the experiment has not yet completed for Chris, so he cannot yet make an objective assessment.
To say "there is no such thing as repetition from a truly objective standpoint" is not true if we allow for abstraction, and we communicate for the most part in abstractions. You could, for example, say that there is objectively a repetition of things which are blue. The only time you cannot have repetition is if you define something enumeratively (i.e. an exhaustive extensional definition) to have only one instance. Confusion might arise if we are inconsistent with the scope in which we ascribe objectivity. Ultimately, whenever somebody states that something is objective, this is necessarily a subjective assessment with many contingencies. However, this does not mean that we cannot talk about something being objective, because we can narrow the scope by assuming various axioms, such as that we have commonality in our language.
I don't really see that there is much of a difference between the subjectivist and objectivist views. We can state the problem you've provided in frequentialist terms and stil conclude that Chris was correct to say 1/10.
Given an infinite number of trials in which the only information which is guaranteed to stay the same is that which Chris is given, the relative frequency of x=5 would be 1/10.
Both Bob and Chris are correct.
Probability is contingent upon given information.
If you create a hypothetical scenario in which you stipulate that some relative frequency distribution exists and that the experiment can only be performed once, then it is valid. Beyond mere stipulation, though, how are we to determine a relative frequency distribution if we have never performed the experiment before?