If trying to focus on Indiana, our asinine hunting laws pretty much make .444 out (case length) and the concept of a .43 Indy (considering the time involved in pressure testing, load developments, extensive hand loading steps to make .43 Indy brass) would be better served in getting good with, handloading for and using .44 Mag at distance. Find low BC rounds and move on with life (or keep writing the DNR and tell them that their restrictions are not rooted in logic). .44 does do well in rifle length barrels with slow powders (akin to using 300MP in .357 mag: terrible and not suggested in a pistol, awesome in a rifle and nips at .30-30 ballistics with just powder and no changes to chamber or components).
Given the relative strength of the H&R/NEF/Henry design and that it comes in .45-70 and some rifle calibers, I'd imagine .444 reaming would be okay in terms of technical strength. You're running into the same ballistic coefficient (bullet is too stubby for long distance) issues with .444 as .44, but then again, I've personally seen a guy take a doe at almost 200 yards with hot handloaded Hornady FTX .44 mag. You're adding complexity, another caliber, load development even with .444 when you can do what you need in Indiana with a .44 mag or hotrodding a .357. Heck, with good powder and crossed fingers, your .43 Indy concept is nothing more than a hot .44 mag.