Sometime ago I wrote a little bit about making this brass from .303 British on another forum.  Let me see if I can remember.  First a .308FL die with the expander ball removed, then a 7-08 die with the expander ball removed, watching the location of the shoulder you made, though that really don't matter since headspace is on the rim, it'll fireform out to your chamber if it winds up a little short to the shoulder, then into your 6x53R die.  The .308 and 7-08 sizing die bodies can be bought as parts from Lee.  I can't recall the trimming that's needed or if it would be best to do that before or after.  I do know they need the neck and shoulder annealed after forming.  Not before, you might run into a problem of collapsing shoulders.  I ran into that when I was making 6.5 Japanese from .308.  Use Imperial Sizing Die Wax and you should have no problems, even with a small press.  I was just thinking back, I posted the instructions to making that on the 1990s internet.  I was going to college then and had found a copy of Nonte's Home Guide to Cartridge Conversions in the library.  That book was so old the .308 hadn't even come about yet.  It didn't get checked out much over the years!

Buffalo has Bertram brass with proper headstamps here => (You need to Login to view media files and links)

And 'formed from .303 here => (You need to Login to view media files and links)

Unfortunately, both out of stock. I bought some of the Bertram brass and have formed from .303 as well to feed my 5 (really? sheesh) Dutch rifles. The forming is not too tedious but be sure to anneal the necks or they will split in one or two firings. FWIW, I annealed the Bertrams as well as they sometimes have a spotted reputation.

Holler back if you need any help. I would be happy to assist and I am sure others would as well.

A friend's wife gave me a brand new set of RCBS forming dies when her husband passed away.  They are made to form .303 Brit into the Dutch or Rumanian Mannlicher & work as intended.

FWIW, the 6.5x53R was the preferred 800 to 1200 yard "match rifle" cartridge until it was ruled off the field by a requirement that the game could only be played with rifles chambered for the 7.62 NATO cartridge case.  With a proper barrel, the 6.5 cartridge will still play with the big boys.

The Mannlicher action, often converted to single shot, was preferred because the trigger was much easier to adjust to the minimum spec than was the Mauser.  Rules for the Match Rifle were the same as for Creedmoor; 10 lbs & 3 lb trigger pull.