Jed, except for the caliber, your new cartridge is somewhat similar to the Sharps 40/90 bottleneck. Case length of the 40/90 is 2 5/8. Also the 40/90 has a longer neck than your cartridge; it's about 0.6" long as I recall, and it looks like your 375 neck is closer to about 0.3" long.
I think there is such a cartridge as the 38/90 Sharps, but I don't know if it was a bottleneck or straight.
I shoot my 40/90 BN mostly with Goex Fg and with duplex loads (RL-7 and Goex Fg), but I've also been playing with 5744 and Trail Boss, mostly with a 320 gr LFN mould from LBT.