Hello homemadeDR,
since a long time in Europe an excellent rimmed deer cartridge with low pressure is the 9x57R but also for stags,
boars and moose.The standard bullet weight is 247 grains.
If you don`t get original cartridges from Browning, you can use modern cases with boxer primer like 8x57 IRS and bring it to 9,0 mm = .350 inch.
kind regards Johann