You should be able to. The bolt face will be the right diameter, and I assume the magazine length will be the same.
The thing that might need work is the feed lips on the magazine/bottom of action. They may need to be opened up as the RUM is a fatter cartridge.
I am not sure what a window style box is. When I had my 30/06 rebarreled 7mm mag the gunsmith gave me a new magazine line which was straight sided, it did not reduce in width at the top, and it had a cutout in the side. That might be what it is.
I think that most of the major die makers would make dies they might call them the 338-300 Ultramag. Get in touch with Shawn Carlock (Defensive Edge), he will probably have some in stock.
The window style box is the magazine box, it is very cheap, instead of being solid metal, it has "windows" cut in it and is a little more bowed to accommodate the larger rounds such as the ultra mags. if you are doing an edge, you should do the wyatt magazine box to gain the tenth inch over the stock box. at that point it works, but some people have just gone the single shot route due to the length they wanted with the 300 smk