Shouldn't 'shall' be 'would' to agree with 'could'?

No, they are different clauses. The verbs don't need to agree.

The first "We shall be obliged" indicates certainty. The second, "if you could look into this matter" indicates politeness (the verb doesn't indicate certainty).

That being said, this sentence sounds poorly written to this native speaker from the United States.

Thanks, Maxdancona.

How would you phrase the text?

You've already asked this question.

I refer to Mr Dancona's missive of the 4th inst., remonstrate his parochial contention and assure him that the sentence is in perfect concordance with Victorian English c. 1840.

Forsooth!

I refer to Mr Dancona's letter dated the 4th inst., remonstrate his parochial contention and assure him that the sentence is in accordance with Victorian English c. 1840.