There is only one area where you might get into serious trouble: the case neck (OK..it's a straight case and doesn't really have a "neck", but we'll call the area where the bullet seats the neck). That 375 brass is thicker at the neck area and the bullets usually used in the 38/55 are a bit larger in diameter than the bullets used in the .375, so there may not be enough room for a clean release in that thick brass. I mesured some 38/55's just over the seated bullet, and came up with .394"(there is a .379" bullet seated in these rounds)...on some .375's, I came up with .400"(these have a .375" bullet seated in them). Seating .359" bullets in the .375cases would bring the outside diameter to .404".
Even modern made .38/55's seem to be using the .378-.379" bore diameters of the old guns (although some old ones will run as big as .381"). Jacketed bullets may work when .003" too small, but lead kind of resents it.
IN short...you could probably do it, but you'll have to measure a bit to find out. If there is enough room for the bullet to be released with that thick brass, it would be safe enough. The combination of a tight chember and a fat case may allow it to chamber, but have no room for the case to expand a bit and let the bullet release.
Have read about converting H&R toppers to .444 (and to 30-40 when starting with a 30-30 barrel). Pressure is about the same as top 44mag. loads (but the peak of pressure lasts a bit longer and falls off slower), case head is a little larger so the thrust back to the action a little greater. The old topper's are not abundantly strong...believe the conversion could be done, but would be cautious in doing it...just a quick sketch of the numbers looks like a 10-12% increase in thrust to the action.
In case you wonder, I've seen a couple of crack-barrel shot guns that have let go (the owner's fault). What seems to happen is that (1) the action cracks at the angle formed at the side of the flate breech/reviever (2) the locking lug pin shears (hard to tell...not all parts get found) (3) the gun unlocks violently...this may be a good thing, the debris gets sent out at an upward angle and missed one guy's face...did pepper his forehead).