If you just want to go as close as you can to factory ballistics, but avoid the use of factory jacketed bullets, you need the hardest alloy you can get, gas checks, and a die to seat them on the bullet. A possible alternative to all of this would be an ungrooved, paper patched bullet which starts out about a hundredth of an inch undersized to allow for the paper.
But if you just want a bullet for casual amusement or use on game from small deer downwards, you can miss out on some of that velocity in the interests of cheapness and convenience, and you may find bullet hardness, which you can now afford to reduce, is not desirable.
One thing you need to do is slug and measure your bore, or push a bullet into the rifling and see if it leaves any spaces for light to leak past in the lands. If this is the case, there is a good chance that a moderately soft bullet (say lead plus 10% tin) will expand to seal the bore under the first blow of the powder gases. With a very hard bullet, including antimony or type metal, the bullet may not expand. What then happens is that hot gases escape through the lands all the way to the muzzle. They blast atomised bullet metal ahead of the bullet, which irons them into the hot bore. Hard alloys may cause more bore leading than softer ones.
Another point is expansion on game. At a velocity which causes a flat nosed or hollow lead-and-tin bullet to mushroom, a very hard one is likely to have the mushrooming lobes break off, leaving a lightened bullet of no more than the original diameter. The scattering fragments have energy too, and may be more destructive on the animal, or less.
The .375 rifle has less case neck length than most rifles designed around lead bullets. Lubed grooves undovered by brass are no more than a nuisance if the throat of the rifle admits them, but you need to make sure they don't pick up dirt, especially abrasive dirt. Lube grooves exposed in the powder space are more harmful. With the lube burned or melted away, they leave exposed lead contacting the bore. I think, though, you should have no trouble getting velocities in the 188 ft./sec. class without stripping the rifling.
It is worth reflecting on why cast bullets may not fly as true as jacketed or other swaged ones. If you imagine a bullet with its centre of mass a thousandth of an inch closer to one side than the other, that centre of mass is actually travelling down the barrel in a spiral path, with pitch equal to the rifling twist, and diameter .002in. But Newton's First Law says that a body remains at rest or in a state of uniform straight-line motion unless it is acted upon by some external force. When the bullet exits, it flies off in a straight line,
in the direction the end of that spiral was pointing. Now, the .375 H&H has a faster twist than most of the black powder or formerly black powder rifles which usually use cast bullets. It will be less accurate unless you can reduce the width of that bore spiral, by quality of components and careful loading to maintain bullet straightness. In particular if you can obtain .375 in. swaged lead bullets, they will probably be more accurate than cast.
There is actually no reason why the .375 H&H bullet needs a meplat at all. If you are interested enough to buy a fairly expensive mould (and you don't need to reduce your .375 jacketed consumption much to pay for one), you will find plenty in
NEI Handtools, Inc .
Weighing fibrous filler material to the grain sounds like a pretty exasperating business. I don't know what is in cigarette filter tips in the US (I don't mean in cigarettes, but in tips for rolling your own) but in the UK it is accurately measured, lengthwise-oriented fibrous material which I believe to be kapok. It could be cotton, but the main thing is that it burns but doesn't melt and smear, and comes out as low-litter, self-extinguished short fibres.