I think there are two kinds of propositions. One is purely mathematical and the other is meta mathematical. When I write former one A, the latter one is something like $\text{prov}_T(G(A))$, where $G(A)$ is the Gödel number of $A$, and this states 'T can prove A'.
According to incomplete theorem, if a formal system T is consistent, T can't prove its consistency.
However, what if we confine propositions? Can't T prove its consistency only for purely mathematical propositions?
Of course, for rigorous discussion I have to define the two kind of propositions rigorously, but I want to know there is any attempt of this sort.
Actually, the Godel sentence is purely mathematical - it's constructed out of the language of arithmetic ($+, \times, 0, 1, <$) and the usual logical symbols (Boolean connectives, parentheses, $=$, and quantifiers). It has a metamathematical interpretation, but that doesn't make it non-mathematical.
In fact, we can do better: by the MRDP theorem, we can replace the Godel sentence with a Diophantine equation! If $T$ is a "reasonable" theory, then there is some Diophantine equation $\mathcal{E}$ such that $\mathcal{E}$ has no solutions but $T$ doesn't prove that $\mathcal{E}$ has no solutions. And it doesn't get much more mathematical than Diophantine equations!
So the premise of this question is flawed. (Incidentally, there are further difficulties down the road. The above is the analogue of Godel's first incompleteness theorem. You asked about the second. This, however, is a much subtler question: what does it mean to be consistent in a restricted class of propositions? If $T$ is inconsistent at all, then $T$ proves all sentences of every form (via proof by contradiction). Presumably you want to restrict also the sentences which we may use in proofs. But now the onus is on you to pin down exactly what sentences are legal, here.)