Conventional wisdom is going to say build the Remington bolt action. A Hart barrel can be a very good choice too. Doesn't have to stay a .25-06 either...
But - as both a Highpower shooter and a military guy - I've seen too much fine shooting done with the good old M-14/M-1A and M1 Garand to discount that possibility. There are obstacles to overcome - scoping it properly can be tricky - then getting a stock with a high enough comb to use the scope well can take a little work too. But doing so, you'd be "doing it your way" for sure.
I'd go the bolt action route with a custom barrel. It's tried and true - works like a charm and there's guys all over who can build you an superb long-range shooter for relatively low bucks on a Remington platform. It's simple and it works.
The M1 though - one of those as a long-range hunter - now that would be something to write an article about...