The K31 has a very smooth action & is a delight to shoot. It is chambered for the 7.5x55 Swiss not the 6.5x55 Swedish (two different countries, but often confused!!).
As to the .303 British, yes the mil-surp ammo is getting a bit harder to find, & most of what I've seen recently is the Pakistani-made stuff which should be avoided at any price. The last I bought was South African, which is pretty decent, & some Jugoslavian which shoots better in my No.4. I have also bought, some years ago, some Sellier & Bellot 174gn FMJ ammo in a plain blue box that shot very well & is boxer primer, whereas the S.A. & Jugo ammo are both berdan-primed. Prvi-Partisan make some reasonably priced .303 ammo in both FMJ & SP, as do Sellier & Bellot. For the most part, I reload my .303 ammo & it is no more difficult, or expensive, than most other calibers. I was lucky enough to buy are rather large quantity of pulled 7.62x54R 'heavy' bullets for cheap off GunBroker a couple of years ago & have been using those mostly, but have also found the bulk Remington 180gn SP bullets shoot well, too. I have a good supply of boxer-primed S&B & Federal cases, but am also set-up to reload my berdan-primed South African & Jugoslavian cases, too (they use the 5.5mm diameter primer as opposed to the larger British one). When reloading for the .303 British, it is probably more critical than with most calibers to neck-size as much as possible, as the chambers are quite 'generous' & constant FLS will result in case seperations & a short case life.