I would recommend the 416 Rigby in a CZ 550 Magnum, with the field grade stock they run just over 1k brand new. A robust 1x4 scope if you want one. Then spend a bit of money with a reputable smith getting the action trued and cross bolts put in. Spend the $2000 you have left over on ammunition - either handloads or store boughts and practice, practice, practice...
IMO the various 416's and 458's will all do the job if you do yours. I bought the Rigby because it is a classic low pressure cartridge that one can load way up well past the factory ammo if one wished to and in all honesty because the Rigby - like the 375 H&H has panache and history. I'd pm BUM (big ugly man) on this site and ask him about his actual experience in Africa and his 416.
I do own a 416 Rigby in a CZ 550, it's a very nice rifle and I thought good value for what it cost me brand new, but it's a whole other world on the recoil side. I consider my 338 win mag a pussy cat and this took some getting used to, hence practice, practice, practice. I'm just passing my first 100 rounds and am starting to get fairly confident on shot placement both on the bench and standing, sitting, etc.
http://w.reloadbench.com/cartridges/416r.html
.416 Rigby
Introduced by the British firm of John Rigby & Co. in 1911, the .416 Rigby is one of the world's great old classic cartridges. Even so, Rigby's .416 did not enjoy the popularity it deserved among American hunters until affordable rifles chambered for it became available during the late 1980's. During 1988, Kimber announced its Model 89 African rifle in .416 Rigby, and Ruger followed suit with its Model 77 Magnum a year later. The author used the first Kimber rifle built in this caliber on safari in Zambia during 1988, and found it and its cartridge to be fine performers on large and potentially dangerous African Game.
The original British manufactured Kynoch factory load contained Cordite, a smokeless propellant which was reputed to be quite sensitive to extremes in temperature commonly encountered in tropical climates. For this reason, maximum chamber pressures for the .416 Rigby was held at 17 long tons or 38,080 pounds per square inch. Muzzle velocity with a 400 grain bullet was rated at 2375 fps. The A-Square and Federal factory loads presently available are loaded to similar chamber pressures and velocity levels.
The .416 Rigby is an excellent choice for hunting the largest African and Alaskan big game. It delivers a mighty blow up close, shoots relatively flat, hits hard at long ranges, and the high sectional density of its 400 grain bullet enables it to penetrate deeply. For maximum penetration on elephant and Cape Buffalo, bullets of solid construction such as the Hornady, the Speer African Grand Slam, the Barnes Super Solid, and the A-Square Monolithic Solid are powerful medicines. Excellent choices in expanding bullets of controlled expansion design for the .416 Rigby are the Barnes X-Bullet, the A-Square Dead Tough, the Speer African Grand Slam, and the Swift A-Frame. Due to the extreme large powder capacity of the .416 Rigby, it should be loaded with magnum primers and powders that fill its case to at least 90 percent charge density.