If you can, try the action of a 'pre - WW2' Mannlicher Schönauer before you decide. There's nothing else like it.

The M1903 was chambered for (readily available) 6.5X54MS. The M1905 (9X56MS). M1908 (8X56MS), and M1910 (9.5X57MS) are essentially for handloaders (a worthwhile pursuit), while the M1924 was chambered in .30-'06. The 'M1925' was available in several chamberings, including 7X64.

The 'factory take down system' is a dandy bit of kit. I have one in 9.5X57 (M1910), a.k.a..375 Nitro Express Rimless 2.25", with its original fitted case.

Within your budget of 5k, you may even find a nice cased example:

Cased M1903 (6.5X54) Mannlicher Schönauer Takedown

If you have a 'custom build' done, you may also consider using an action from a '1903 Greek' military Mannlicher Schönauer as they should be somewhat readily available. The 'Greek' MS (Y1903, Y1903 - 14, 1927, 1930) and the MS sporting rifles, carbines, all have shared roots in the M1900 and use the same incredibly smooth Mannlicher Schönauer action and rotary magazine.

Best o' luck, and keep us posted.