The question of the general purpose big game rifle can be answered in a number of ways. As far as I'm concerned the ideal all around big game rifle cartridges are the .30/06 for North American hunting and the .375 H&H for around the world. These cartridges might not be your ideal, and for North American hunting I don't know of any cartridges that can do the job decidedly better than either the .270 or the .308. If your custom rifle is to be exactly the way you want it, you know better than anyone else what your ideal cartridge is.
Obviously you are a handloader, or you would not be interested in the Improved cartridges. The .280 AI has much to offer, but not so much more than the .270. Conversely a .308 can be made into a very light , wonderfully accurate, compact carbine that gives up little to the faster rounds. Modern brass tends to have a short life, and that should be considered before choosing a wildcat, even one that is as simple to form as the AI versions of factory rounds.
Worry less about the cartridge your rifle will be chambered for and more about the components that are to go into it, without loosing sight of what the rifle is for. Which action will the rifle be built around? Will the trigger be factory or aftermarket. How heavy should it be? Will it be fully adjustable for weight, creep, and over travel? Is the magazine box long enough for the length of your cartridges? How long a lead will the chamber have? Will you have the barrel chambered for a specific bullet seated to a specific depth? Don't choose a barrel contour so heavy that the rifle feels clubby. Conversely, don't choose a very light contour unless the barrel is to be short. How will this choice effect the balance and point-ability of the rifle? Make an informed decision over a CM barrel or a SS barrel. What rate of twist is the most suitable for the bullets you are likely to shoot. Don't try to make the rifle so light that recoil becomes unmanageable. When purchasing the scope, do so with an idea of the range at which you are most inclined to shoot. Don't over scope the rifle, either by physical size or by magnification. Get the best mounts you can find. Will you choose iron sights, if so which ones? Choose you stock material with care, and consider the clothing you might be wearing before deciding on the length of pull. What type of bedding is the most suitable for your stock material? Where will the sling swivels be placed? Will that placement allow the muzzle to be carried low enough when slung? If on the forend could the swivel placement result in a cut hand?
Planning a custom rifle can be a joy or a nightmare, its better to enjoy it.