I have a 700 Classic in 350 RM, did some extensive load development back a few years ago. I never ran into your particular issue, but I did have some die issues. One of my issues was a FL die that was setting the shoulder back a whole bunch, which I "fixed" by buying a custom shellholder from Redding.
That isn't your problem. I'm assuming that because your rifle shoots factory ammunition just fine, your rifle is not the issue. It's been my experience that R-P brass works pretty well in the 350 RM, so brass is not the issue. You can chamber the ammo you make, so your dies are not the issue.
Unfortunately, without any more information, I can only conclude that your ammo is the issue. The 350 RM is not hard to load for, but there certainly are some nuances that one needs to be aware of. The 2 main manufactures of rifles in this calibers were Remington and Ruger. They are different.
What rifle do you have? And because there were production changes, what twist rate is the rifling. In an attempt to "correct" issues, the twist became faster and faster as time progressed. It started at 1:16 and the last rifles were 1:12 or 1:10.
The next consideration is the bullet, because in the 350 RM this can make a huge difference, depending of course on the brand of rifle you own.
Another unknown is what OAL you are seating the bullet you are using. Deep seating to accommodate the very short Remington magazine precludes the use of some bullets and changes loading data. Some data is for Ruger rifles, not the same as a Remington. Maximum OAL is just under 3" in a Ruger, and in a Remington it needs to be under 2.8", not much under, but under just the same if you want your ammo to fit and feed properly.
It would be nice to know what powder you are using, and of course how much.
Varget would be my powder of choice, especially with a 200 gr bullet. My bullet of choice is a 200 gr FTX, but whitetails are my game of choice. The 200 FTX may not be the best choice for moose, especially if the distance is fairly personal. The FTX expands at the lower end of the velocity range, having been developed for the 35 Remington, with a tubular magazine like the Marlin 336, also a great bullet for that application. The 350 RM also is nearly perfect for shooting cast, if you are so inclined. The upper limit for the FTX is about 2800 fps, and you can get pretty close to that with handloads.
If I were to guess, you are probably using a 250 gr bullet, seated fairly deeply so it fits in your magazine. In all likelihood you are also compressing the powder slightly. You are probably using data for a longer OAL and you are compressing a powder that doesn't particularly like to be compressed and are experiencing excessive pressure. Of course chamber and reloading die issues may be the problem, but I'm thinking that they are not.
Again, as stated above, a picture of a factory fired case, and one of you handloaded ones would be invaluable. A quick and dirty pressure analysis on a belted magnum is pretty easy, just ahead of the belt there is a short distance that is still within the web of the brass. When fired this area will not usually touch the chamber, unless pressure is fairly high or the chamber is fairly tight. Depending on your die, this area can be sized if your die is somewhat on the small size. If you look at the picture I've uploaded, it shows this area, smallest on the left and largest on the right. There are other factors, but this is what you'd see if you increased pressure of your loads, normal or starting load on the right, approaching max on the left.
I hope this helps, if not, could you add a few details and pictures.
Nitro
https://aussiehunter.org/shooting/calibres/5690-2/350-remington-magnum/
http://35cal.com/35bullet_study/35bullet_study1.html
