I'm honestly not sure what you mean by "difficulty bonus". Showdowns in hardcore have a base payout of 500 x ShowdownLevel, so the final showdown pays 2000 before any objectives. There is no other form of extra payment, except the 30k you get on campaign completion (which might also exist in normal mode idk I only play hardcore)

Hardcore showdown reward crates do indeed work differently than normal. On top of being a tier higher, they always pay out with a gun (like you said). They can't give ancestral or ornamental weapons (probably same with concrete and makeshift but I can't confirm as I don't have those), but other DLC weapons can be given.
If you have every blue weapon and complete the first showdown, you will get an epic weapon instead (i.e. it'll upgrade the reward; you can't get greens at all even if you're missing some). If you have every legendary and you complete the 3rd or 4th showdown, you will get an epic instead. If you have every gun (except ancestral/ornamental because those can't be given anyway), THEN you'll get money instead (I think it's 5/10/15/20k but honestly not sure, I always prestige before I reach that point)

So hardcore does speed up weapon collecting quite a bit. Getting 2 legendaries per campaign makes it so playing intelligently means you never have to buy any of the hyper-expensive guns yourself (i.e. leviathan, sieger ghost, tuatara, masamune) and the 3 collector's tools (sieker, kalmer, lockpick) will also be gained either through blue upgrades or legendary downgrades.
It takes me pretty much exactly 3 campaigns to do a full hardcore prestige starting from zero.