It is possible to craft all weaponsmithing weapons (Thunder, Lionheart Blade, Lunar Crescent and their upgrades and 1-handed equivalents) even if you're not specialized in any type of weaponsmithing. The weapons have a Master Sword/Axe/Macesmith requirement needed for equipping them, but while I wasn't able to find trustworthy evidence of this being the case, I'm very confident you need the specialization to learn the recipes and craft them in the first place.
If i recall correctly, you needed the specialization to learn the recipe, but once learned you had it even if you changed to another specialization, despite not being able to wield your mighty weapon.
It is possible to craft all weaponsmithing weapons (Thunder, Lionheart Blade, Lunar Crescent and their upgrades and 1-handed equivalents) even if you're not specialized in any type of weaponsmithing. The weapons have a Master Sword/Axe/Macesmith requirement needed for equipping them, but while I wasn't able to find trustworthy evidence of this being the case, I'm very confident you need the specialization to learn the recipes and craft them in the first place.