The .375 Flanged Nitro 2 1/2" is very similar to our .38-72 cartridge, although it seems to have been loaded to a bit higher velocities than the .38-72 cartridge.
I've seen some target rifles built on the old Martini action, mostly with heavy round barrels, although I've often wondered how a large octagon would look on this action? The pistol gripped variations had a custom built lever, which was bent and flattened to hug the front edge of the pistol grip, with a ball or button on the bottom extending slightly past the bottom of the grip. This makes a sleek design, and works great too.
Most of these have had hard rubber buttplates, or checkered steel flat buttplates. Forearms were small schnabel style, although one I saw had a beavertail type that felt great, but looked out of place on the action type. Finely checkered delxue wood is common, and looks at home on these guns. Sights seem to be the biggest problem, as there's no tang to mount target style sights. Some of the Parker Hale, or other aftermarket Martini sights might be a good choice, or even a target style external adjustment scope, if your eyes are less cooperative like mine!