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No I think that this direction is the false one. It's the other way round: If ZFC proves a statement then ZFC proves that the 'coded ZFC proves the coded statement'. If your suggested direction would be true it would cause trouble with Gödels Incompleteness Theorem.
Just wanted to say that the continuum might have any regular uncountable cardinality you want. This is a result by Easton. So the size of $K$ isn't the point
Thank you very much for your answer. There is something I don't understand though: In your third break you write:" ...by Montague people living in $M$ believe that $\{ \phi_{0},...,\phi_{n} \}$ has a model for each $n< \omega$." And now you conclude with compactness that $M$ thinks there is a model of ZFC. But as Sridhar pointed out, ZFC does not prove " for any finite subset of axioms ZFC, there exists a model". Hence $M$ does not believe that $\{ \phi_{0},..,\phi_{n} \}$ has a model for each $n< \omega$ and one cannot derive that $M$ thinks that "there is a model of ZFC". Am I wrong?
