Hello Kenneth Mcmillan,
Personally, I would not select the lever action for African heavy / dangerous game.
Generally speaking, lever action rifles usually tend to be chambered in short, marginal cartridges (low powder capacity/potentially high chamber pressure).
Some Winchester / Browning 1886 actions have been used as the basis for "wildcat" lever action cartridges, with Africa in mind.
However, I would worry about chamber pressure in hot weather, (as well as my PH wondering why on earth I would bother trying to re-invent the wheel).
The Winchester 1895 lever action was / is chambered in .405 Winchester which, could be argued is a very fine lion cartridge.
I would not know since I have never shot or even seen a lion shot.
That being said, lions are impressive animals when not upset,
I would not want to face an upset one, while I was carrying only a .405 caliber rifle.
The .375 H&H / 300 grain bonded soft, comes to mind as a decent minimum IMO (only slightly more energy but much better penetration potential).
The .450/400 NE. .404 Jeffery, .416 Rigby or, any of the even larger bore Africa proven cartridges above these, including the .500's (not the black powder versions) seem like they'd be so much the better.
The .577 seems a bit much to carry all day (especially at my ripe old age) and by reputation, all of the other ones mentioned above, should smartly crumple the grumpiest lion, (with a reasonably well placed shot).
Regarding other dangerous African animals, in some countries over there, lever action cartridges, "wildcats" included, typically do not meet the legal ballistic requirements for certain species.
If you are serious about hunting African dangerous game, just buy yourself a sturdy bolt action, such as the Winchester Model 70 or CZ 550 magnum, in .375 H&H, then learn to shoot it exceptionally well (about twice the recoil of a .30-06 with heavy bullets).
If you are already quite comfortable with that much recoil and are considering truly heavy game (you mentioned elephant and rhino) then, I suggest a .458 Lott.
With 500 grain monolithic solids, it is a penetrator for sure - recoil is quite "sporting' though, LOL.
Parting Shot as it were:
I feel you would be better off with even a single shot, chambered in a proven caliber than, you would with some less powerful lever action, no matter how many cartridges it held.
Cheers,
Velo Dog.