A restrictive relative clause may use either that or the appproriate wh- form as relativizer—or, in some circumstances, no relativizer. None is "better" or "worse" than the others: it's a free stylistic choice.
Consequently, you may write any of the following:
Catholics were aligned with Southerners in the Democratic Party at midcentury, a coalition which was exceedingly unfavorable to Catholics on the issue of public aid.
Catholics were aligned with Southerners in the Democratic Party at midcentury, a coalition that was exceedingly unfavorable to Catholics on the issue of public aid.
Catholics were aligned with Southerners in the Democratic Party at midcentury, a coalition ∅ exceedingly unfavorable to Catholics on the issue of public aid.
Note, by the way, that what matters here is the relationship between the relative clause and the term, coalition, which it modifies; the syntactic role of coalition within the sentence is entirely irrelevant. In fact coalition is not a modifier, restrictive or non-restrictive, on Democratic party: it's an appositive which restates the entire main clause.
