The .444 Marlin or .38/55 would be a great choice,as would the .375, a necked down .444.

If building a .444, I suggest you get a faster twist in the 20" to 24" range. The typical rifle twist for the .44 is 38", which will not normally stabilize .430" bullets over 300gr.

If going .38/55, a .375 Winchester chambering might be better as it has stronger brass and components are quite readily aquired. .38/55 brass is typically .050" longer than .375 or .30/30 brass. I use blown out .30/30 brass in my Winchester .375.

I'm with Peter on ctg. choice - as to a .45/70, although a .444 with 20" twist for 350gr. to 400gr. bullets has a great deal of attraction for me. It should do about 1,700fps with a 400gr. at under 40,000PSI, maybe 1,850fps with a 350.

You would be hard pressed to get a 200gr. bullet at 2,400fps from the .38/55 (or .375Win.) case at safe pressures for the double, I think - about 2,300fps is about tops andthat would be over 50,000PSI. Lee's handloading book has data for AA1680 pushing 220gr. to 2,400fps from 24" bls.(not specified). I tired them in my .375Winch., which normally operates around 55,000PSI & felt they were way too hot. With this small capacity case, it takes a lot of increased pressure to get just a few more fps. More capacity would be better in that calibre.

A .375JDJ (necked .444) chambering on 24" tubes might deliver 2,250fps with a 270gr., and the new .375", 225gr. Spire point by Hornady, a very heavily jacketed bullet should get around 2,550fps at reasonable pressures for the double - say, 45,000PSI. These ballistics are capable in a 9.3x57 in a m46 Husky, which operates in that pressure range with just a tich less capacity than the .375JDJ.
This bullet worked splendidly in 9.3x57's bullet testing on jugs and boards as well as Steer heads. It produced equally good results on 2 deer for him, with almost full length penetration and excellent expansion. A good choice for a fun gun, in my opinon.