Oh, they really do.
It is changing rapidly as the Vatican is actually concerned about stuff that is the Gospel these days, things like good Catholics being supposed to adhere to hospitality, sharing and compassion toward less fortunate people (such as, for instance, refugees), a point that many of the current Italian mainstream politicians (MOSTLY of the populist right) vehemently deny on a identity politics nationalistic basis; the conundrum is that identity politics nationalism in Italy needs to pay lip service to Catholic identity (supposedly threatened by all these dark-skinned persons trying to reach safe haven in Europe). So the right cannot really play anti-clericalism, indeed there's a faction there that aligns closely with the most conservative groups on the Church on things like equal marriage, reproductive rights, divorce, abortion, and similar issues. They actually espouse a far more narrow and patriarchal view than anything the Church ever officially approved, but their views on family are quite in synch. The centre-left opposes this (and often saw its Catholic and secularist wings clash) but is traditionally very keen on working with the Church on social issues and shares at least notionally some ingrained Catholic ideals about solidarity and welfare.
As immigration and identity increasingly become centerpieces of public debate, of course, all these tensions explode and the traditional alignments of the Church on the Right become less and less tenable as the Vatican really insists that Salvini's rhetoric is plain un-Christian. The result so far is alienating a lot of right-wingers from the Church, or from the current Papacy at least (with a lot of internal unease about Pope Francis emerging) but in the long run, the Italian populist right cannot afford to just turn anti-clerical outright. The secular left also cannot provide an alternative interlocutor for the Church since they a) are weaker anyway b) while they cooperate with Catholics on social issues, they disagree on almost everything else, and despise some significant Church interests (private schools being particularly contentious). So, the Church has a lot clout but cannot use it effectively for internal divisions.
As a side note, even Neo-Fascists are divided in "Catholic" and "Secular" camps, with two distinct political movements (the secular one being by far the most numerically significant right now); the Church disavows both and both show sympathy to Salvini, but in some places, the "Catholic" neo-fascist movement has connections to local sympathetic clergy.