Annex Mexico, Canada, Cuba and the Philippines. That’s probably the most Catholic the US could plausibly get
Some notes to this:
-- A USA that wants Canada should in most cases be a USA that gains it early. The relevant factor here is gaining Quebec, but even supposing the Maritimes stay with Britain (increasing the relative weight of the addition of Quebecois Catholics), holding historical 'Canada' means the USA will almost inevitably gain domination over the OTL Canadian west as well. Any additional ATL settlers going there will likely not be Catholic.
-- Arguably, a conquest of Cuba in the context of an early Spanish-American War (say, a decade after this ATL USA has annexed all of Mexico) could afterwards see US filibusters conquering the Dominican Republic, Haiti, Puerto Rico and Nicaragua (to name the most obvious candidates).
-- Conceivably, the annexation of all these Catholics could see some sort of discontent arise whereby a lot of rural, Protestant America eventually secedes from this ATL USA (one could realistically argue that a much bigger USA could very well be more decentralist, making secession a more acceptable and less preventable factor). Ultimately, everything west of the Mississippi and roughly north of the OTL Missouri Compromise Line (except California) secedes, including the (also primarily Protestant) Canadian west.
-- Presuming for a moment that this is a rather 'evil' USA, with the caste system I suggested above, and where north-eastern industrialists have opted to compromise with southern slavocrats to create an apartheid-like system that retains (quasi-)slavery, Britain might support such a secession movement, making it extra realistic. Secession of this (mostly agrarian, mostly white) region is furthermore realistic because it's an area that neither profits from the caste system, nor likes being part of a Union that holds so many catholics and non-whites.
-- Since California stays with the USA, a powerful interest in the Pacific is retained. This eventually gets the USA the Philippines. Even more Catholics!
-- Yet Hawaii... er.... 'the Sandwich Islands', that is, get nabbed up by Britain.
-- During a series of Banana Wars, and seeking to build a Panama Canal, this USA eventually annexes all of Central America, adding yet more Catholic denizens.
-- You end up with a USA that has as many Catholics as possible, while an area that was very Protestant has been cut off. This is probably the highest number of Catholics you can get, both in absolute and in relative terms.
-- When the horrid caste system eventually blessedly collapses, Catholics (and non-anglos, at that) will be an outright majority, bringing us to the prallel of post-Apartheid South Africa.
-- It's not even unthinkable that, when the system fails, the North-East at least tries to bail out via its own secession. An American analogue to the OTL South African white 'Volksstaat' suggestions would be more realistic than the actual OTL Volksstaat idea, considering the rather white nature of the furthest North-East, and its greater population. Supposing Catholics in NYC and Boston balk at this and the now-Catholic USA retains them, you'd see Maine, New Hampshire, Vermont, and the north of upstane New York forming this secessionist state, cutting off yet more Protestants from this ATL USA.
That's about it. I cannot for the life of me imagine a USA, with a post-independence POD, that would be significantly more Catholic than what I've sketched out here. It would look something like this (very crudely drawn, internal borders unedited, using the Canada coplour for the north-western secessionist state and grey for the north-eastern one):