In a Remington 700 I would build a 338 RUM. I know Remington makes them but I don't like that big of rim on a 700 bolt face. I had one of the factory rifles and I could not get any speed at all out of it without massive pressure signs and the accuracy was not acceptable. I wouldn't even stay supersonic to a mile. You might consider a dedicated 300 RUM for the 230 Berger. I did one a year or two ago that is doing incredible things at long range and it's nipping at the 300 grain Lapua in energy. 230 grain bullets at 3200fps. Brass is seasonal but if you are patient it can be had.
If you really want a 338 Lapua I would sell the 700 and pick up a Stiller Tac 338 action. It makes for a very sweet 338 Lapua. I have one and have built many for others. Mine has a PacNor barrel on it and it is a tack driver if there ever was one. Keep an eye on the new Curtis action that Joel Russo is involved with. It's a 3 lug action and very well thought out.
http://www.curtiscustom.com/product/vector/ They also have a budget model for regular calibers showing for $750.
