Peridot doesn't have a weapon.

As far as we can tell, Alexandrite can only use other fusions weapons. But considering Obsidian has her own weapon, the reason for this is unknown.

Bloodstone is most likly a real Gem type.

Not all gems manifest a physical weapon.  Peridot has ferrokinesis which allows her to manipulate metal objects.  It's highly likely that other Peridots have this latent ability.  Lapis Lazulis have hydrokinesis allowing them to control water.  Blue Diamond can impose her emotional state on other gems including other diamonds.  Yellow Diamond has her destabilizing bursts that poof any gem in range.  White Diamond has her mind control abilities.

If Alexandrite had her own weapon my guesses would have been a Crossbow Balista.

Well I'm also wondering what Demantoid's weapon could be (Demantoid=an angry wittle green tree from Unleash The Light) considering that Hessonite and Pyrope have weapons and they're Era 1 Gems while Demantoid is an Era 2 Gem (though she probably has no weapon cuz of it being Era 2; same with Peridot; Amethyst has a weapon due to her having enough resources)

I think that Alexandrite's weapon is a spiked flail