I think I found the definitive answer. D&D 3.5 DMG, in the Special Materials section.
There it talks about Mithral and its use in crafting armor. It lists prices for Mithral when used in armor, and makes no mention of Mithral weapon cost at all.
I know, not convincing enough for some.
A little later in the same section though, on Page 285, it talks about Alchemival Silver. This is the silver alloy you lay over a more normal weapon to make it a "Silver" one. Actual, straight silver is far too soft to make a good weapon of any kind.
In that section it mentions that you can't lay Alchemical Silver over non-metal items, and that it "doesn't work" when laid over rare metals such as Cold Iron, Adamantine or Mithral.
If Mithral counted as Silver, why would anyone ever want to lay Alchemical Silver over it?
Mithral is tough and light and capable of fine linking; makes the best mail. It's not for weapons. Possibly a mithral weapon could have the "light" quality, if it normally didn't (or could remove the heavy property), but that's it.
Silver is magic, because it's the part of the moon. You bury souls with silver on their eyes so they can go to the underworld. It is useful against lycanthropes. Weapons of all kinds can be silvered because it's not the metal that does damage, it's the moon-magic that it belongs to.
I don’t recall elves ever calling it that, but the dwarves do several times. It’s essentially a magic metal that is silver but doesn’t tarnish and is light but strong, and can be worked like copper but when set and forged becomes harder and stronger than steel.
I would allow it to hit like both steel and silver. There’s a reason that mithral armor is a magic item.
The closest IRL thing to mithral is intermetallic yttrium silver (YAg, actually chemically bonded to each other, not just an alloy). We were given an extremely clear and explicit accounting of mithril's properties: "It could be beaten like copper, and polished like glass, and the Dwarves could make of it a metal, light yet harder than tempered steel. Its beauty was like to that of common silver, but the beauty of mithril did not tarnish or grow dim." Thing is...it's normally impossible for a metal to exhibit these properties. Even the mentioned titanium cannot be beaten like copper can--in fact, for most metals and alloys, hardness is literally the exact opposite of that, because hard metals specifically resist being hammered out, that's what hardness means.
But intermetallics can break the rules.
YAg hits perfectly, having almost exactly the same color spectrum as silver. In its natural state, it is malleable like copper and can be polished like glass. But--and this is the best part--if you quench-harden it after forging, it develops a multitude of fine separations of metal crystals, which cause the exterior to become extremely hard, and in specific, harder than tempered steel. Not only that, but after being forged this way, YAg apparently has a very beautiful "glittery" surface, as the tiny microcrystalline "cracks" reflect the light in different directions, making it rather striking to behold when you "make of it a metal."
Even better, mithril in the books is notorious for only coming from a small number of mines, and always being in rare supply. It's entirely reasonable that a naturally-occurring lode of intermetallic yttrium silver occurred in Moria, possibly having splintered so other small lodes appear elsewhere. This would explain why it's so incredibly rare, and why the dwarves would want to "delve too greedily and too deep" in the first place.
By comparison, alchemical silver is most likely some kind of electrochemically-plated wash that you put onto a weapon. Coating armor with it is impractical, particularly since it doesn't really confer any benefit for doing so. And since it's electrochemically plated, it can wear off with time and polish/sharpening/etc., hence its benefit may not last forever.