You could probably get Hume to admit to these Leibnizian claims:
Sufficient reasons are evident from terms.
Facts are never evident from terms.
Analyses of facts that are bound to what is sensible yield only facts.
Leibniz adds the PSR to these and gets a cosmological argument, since to break out of the bounds of the sensible is to posit the trans-cosmic. Hume’s whole epistemology requires that analysis be bound to the sensible, which requires some denial of the PSR.
There is probably less distance between these two opinions than at first blush. Hume’s discussion of fundamental causes is not characterized by rejection and denial but by a sort of humility in the face of what can’t be known and a corresponding refusal to talk about it. The springs of natural action are simply unknown to us and should be respected as unknown. But note the first thing Leibniz says about the PSR:
32. [by the PSR] there can be no fact real or existing, no statement true, unless there be a sufficient reason, why it should be so and not otherwise, although these reasons usually cannot be known by us.
Although these reasons usually can’t be known by us. The cosmological argument is an inference from a world largely unknown and perhaps unknowable, and so with certain facts that we can analyse no further. It’s not as if Leibniz’s vision of our epistemological predicament is one where the human mind has cast infinite light and intelligibility in all directions, and then at the far end of a set of Euclidean interences one trumphantly concludes to God and his attributes. The inference to divinity is one that we make from a world that is mostly in the dark, perhaps permanently so. Theism is not a seeing into nature but a judgment about a mostly unseen world.