In all seriousness, while everyone has the right to live their own life as they see fit, your neighbor's big cat collection has no place in the USA. One of the great problems in ecology is the introduction of non-native species. Witness the infestation of pythons in the Florida Everglades. It is estimated that there are over 100,000 pythons living in the Everglades now and they have decimated the wild pig, deer and small mammal population in what used to be a pristine wilderness. This infestation began by people releasing their no longer wanted pet pythons into the Everglades.
I strongly suggest you contact the USFWS and bring your cat problem to their attention. These animals do pose a serious threat to people and livestock. The escape of these apex predators would be costly to both populations. And a nasty mess to clean up, especially if they start breeding.
For your peace of mind, my suggestion is a .375 H&H or .416 CRF rifle loaded with Woodleigh Weldcore softs. You can buy cartridges loaded with Woodleighs from Superior Ammunition. The Kimber Caprivi .375 has a four round magazine. The less expensive Winchester Safari Express is also a good choice and Dakota, at a higher price point, makes a superb rifle as well. If you must have a .458, Kimber does also chamber their Caprivi in .458 Lott. The Lott is, IMO, preferable to the .458 Winchester and will chamber .458 Winchester ammunition in a pinch.
If rapidity of followup shots is an issue that concerns you, consider a Blaser R93 in .375 H&H. The straight pull bolt can be operated very quickly from the shoulder and the interchangeable barrel feature allows you a wide choice of calibers. The newer Blaser R8 can be had in calibers all the way up to .505 Gibbs. However, if you decide on an R8, special order it, specifically deleting the detachable trigger/magazine feature. A detachable trigger/magazine can detach at an inopportune time, rendering the rifle useless. I watched a friend's video of his hog and deer hunt, as he become extremely frustrated on two occasions when the trigger/magazine assembly fell out of his R8. That feature is an ingenious solution to a nonexistent problem. Fortunately, the trigger and magazine on the R93 do not detach.
You will want a lower power scope with a wide field of view, illuminated reticle and good light gathering capability should you have to shoot in low light. Swarovski and Zeiss come to mind, but an excellent scope at a lower price point is the 1.5-6 x42 Meopta, with an illuminated reticle. You may need to use the rifle's express sights at some point, so QD mounts are essential on a DG magazine rifle. I have found EAW's to work well, but they are overly complex and require periodic maintenance. A better choice, IMO, is the Recknagel. I have both EAW and Recknagel mounts on my rifles and prefer the Recknagel by far. Blaser has an excellent QD mount that is proprietary to their R93/R8 series rifles.
I have taken the Big Five and have found the most intense hunting experience of the five to be the leopard and lion. A wounded cat is quite an event, so use enough rifle to put them down quickly with the first shot if at all possible. My lion weighed 450 lbs. I am informed that a large male tiger can approach 600 lbs. I would not suggest a .30/06, .35 Whelen or 9.3x62 for an angry, hungry 600 lb apex predator. Considering a .375 to be your minimum would be prudent. Again, I don't know about tigers, but an angry wounded lion or leopard is unlikely to be turned away during a charge. You must kill it to stop it. The best insurance against a charge is a well placed first shot with a quickly expanding soft that will still hold together and punch through, imparting lots of shock along the way. A fast stepping .375 loaded with a Woodleigh soft will do just that.
From what I have read, tiger hunts in India were either conducted from the backs of elephants or from a machan or high hide. Since you are unlikely to have elephants and beaters at your disposal, consider building several machans in advance at likely spots, near cover and water and with a silent and concealed approach path for your entry. Leopards and lions are opportunistic scavengers, as well as being hunters. They can be baited by placing the bait in an appropriate tree and shot from ground level blinds. I don't know that a tiger can be baited and have read somewhere (Corbett?) that wild tigers only feed on their own fresh kills, though captive tigers must assuredly eat dead meat that is fed to them by their handlers. That may mean that escaped captive tigers can be baited as well, but that is pure supposition. According to the literature, tigers are fond of attacking from behind. That is likely why tiger hunters favored machans instead of ground level blinds, but again, this is conjecture.
For further details, I strongly suggest you obtain a copy of "The Perfect Shot II" by Kevin "Doctari" Robertson and a copy of "Chui" by Lou Hallamore. These excellent books discuss shot placement, hunting techniques, rifles, cartridges, bullets and hunting strategies in depth.
