I've done a number of these projects with both 14s and 17s. They work well.
I completely agree with your choice of .300 Winchester Magnum. Another viable option would be .300 WSM. If you're building a serious rifle for long range target accuracy, these two cartridges would be the top of the heap for that action, with the preference being .300 Winchester. However, a .300 Holland & Holland or a .308 Norma would also be great.
If you're serious, you should spend the bucks and get a premium target barrel. Our Canadian makers, McLennan, Gaillard, Smith and McPhee all make a spectacular, world-class barrel. Expect to pay in the range of $350 for a brand-new stainless match barrel blank and around $150, give or take, to have it chambered, threaded and headspaced.
If this is done right, you should end up with a rifle that is a serious contender. I agree that this project might exceed 50% of the cost of buying a new heavy-barrelled Remington 700 sniper rifle, but the truth is that your (McLennan/Smith/Gaillard/McPhee) barrel alone will be superior to any factory barrel out there. (True, some factory barrels shoot very well, but one of these premium target barrels is almost guaranteed to shoot better.) If you were thinking of a simple hunting rifle (say a Savage), then the factory option would be cheaper and more practical than building one out of a P-14 action with a custom barrel. But if your goal is to build a super-accurate rig with a target barrel, your P-14 will still be cheaper than buying a brand-new Remington 700P, and it should shoot AT LEAST as well.
Let us know how it works out.