First, I wonder exactly which animals you intend to hunt where that the .375 H&H might not handle even if things "went bad." I have hunted in Africa some and know quite a few PH's, and they are just about universal in their sentiment that the .375 is the perfect dangerous-game rifle for the visiting hunter (well, with the possible exception of leopard, where many would suggest something even less powerful). In fact the most experienced PH I know uses a .375 as his back-up rifle when guiding clients for Cape buffalo.
As for the CZ screwing up, I have seen more CZ's in the hands of PH's than any other bolt rifle over the past few years. This partly because they are an affordable controlled-round-feed rifle, but also because they are quite reliable. There are ways to make the trigger more reliable, though most of the PH's I know have had no trouble with their's. And I have never had any trouble with the trigger on my own CZ .416 Rigby in several years of use.
Most PH's will be happy to have you use whatever rifle you want to, but I don't thinka any I know would suggest a double for anything except possibly elephant. This is because a big double is much harder to hit with at anything over fairly close range, when compared to a scoped bolt (or single-shot) and putting the first shot in the right place is by far the most important factor in the whole equation.