Thanks for your email.
Here's something that even experienced gun writers don't know: swaging is the opposite of drawing, because swaging always expands material in a closed die, whereas drawing always reduces material in an open ended die. So what you are looking for is a DRAW DIE (BRD-1-R Bullet Reducing Die). Not a swage die, which is used to create bullets using lead and jacket material.
Also, swaging is pronounced like "paging", with a long A, and not like "wagging" like a puppy's tail. (People often say "swagging" when they mean swaging).
Now you know more than most gun writers, at least about that part of bullet and jacket making!
So drawing down an existing bullet is just as simply as pushing it through the draw die. But there are limits. Usually we try to draw down no more than .006 inches. The reason is that the jacket can "spring back" a bit, making it loose on the lead core. Lead is a "dead" material: you squeeze it, and it stays squozen! In other words, no spring back. If you keep the reduction around .006 or less inches, usually there is no significant effect on the quality of the reduced bullet.
Now, .358 to .348 is .010 inches. So is that too much?
It depends. If the particular bullet jacket is very springy (usually harder alloy or work hardened in making the bullet) then it can create a loose core after drawing that far down. But if it was annealed and not drawn too much further after that step, it might be less springy and work just fine. No way to guess.
But you can send me a few sample bullets of the same brand you plan to use. Same exact bullet, so I will get the same results as you. I can push it through a draw die that is close to .348, maybe a little smaller, and check the core by cutting it in half on a mill. If it is loose, then it won't work and I wasted my time but not your money! If it is snug still, then maybe you'll go ahead and buy a draw die because it will work, and we just proved it.
The idea is to keep you from having to find out the hard way!
All you have to do is send me six sample bullets. I will test and destroy one of them to find out what we need to know. If the draw works with that bullet, then the other five can be used (if you order the die) to adjust the die as we lap it to exact finished bullet size. Actually, since there is some spring back, we lap the die to what we estimate will be about right but on the plus side. Then we draw down a bullet, mike it carefully, and usually then need to lap just a little more and try again. Within five attempts we will have it right on the size you want to the nearest 100 millionths inch. (We've done it before, over the past 50 years, so it's easier for us now.)
We send the bullets back with the die except maybe the one we ruin by taking it apart to check the core.
Oh, and the BRD-1-R fits any standard reloading press, and is $179. Shipping usually is about $14.50 in the USA, including insurance. I recommend getting a 2 ounce bottle of Corbin Swage Lube (CSL-2) which is $8, because it greatly reduces friction and extends die life to almost infinity. (Pretty good case size lube too, and non-dieseling air gun lubricant! We private label it for Boeing Aircraft suppliers for use as a certified safe lube for hydraulic fittings, among other things.)
You can see it and order it on www.SwageDies.com
web store or by email, whatever. I'd say phone it in but we've been so busy since March and all the riots in the cities that it's tough to break through the constant stream of other phone callers! But you are certainly welcome to try the phone.
The BRD-1-R fits a reloading press, and the other BRD-1- dies have -S or -H tacked on the catalog number, because that means they fit our S-press or our Mega Mite or HydroPress models.
I've attached a file that explains the die but it is pretty simple to grasp, unless you are a liberal who can't even understand that they are destroying their own freedom and the future of their children by pushing socialism on America. (I think most shooters are not that self-destructive.)