Not to get into this again but they are not a wizard. They count as a wizard. Therefore they can wear armour (unlike a wizard) but can't take arcane items (like a wizard)
No offense, but you need stop making up your own rules...
Wizarding Hats rules say "The wearer is a Level 1 Wizard". Nothing about "count as" or "cast spells as a Level 1 Wizard" just "they
are a level 1 Wizard", and as such, they are allowed to buy Arcane items, and make dispell attempts, etc, because that what Wizards are allowed to do.
Then the FAQ clarified that they can still use armour (because of course they can, the item would be mostly useless otherwise) by stating that they "aren't actually wizards". (Is this gonna be the next piece of flavour-text that people are gonna get into arguments about?)
So we have two contradicting statements: "Level 1 Wizard" and "aren't actually a Wizard", but the latter is imo more of a flavour text than a rule - of course that Ogre Tyrant isn't actually a Wizard Hagrid, but he follows all the rules for them thanks to his Wizarding Hat, as far as the game, rules and interactions are concerned.
There's also a seemingly subjective picking-and-choosing of what goes and doesn't in this weird Wizard-hybrid-combo;
- Can they cast spells? Yes, they are a wizard.
- Can they dispell spells? Yes, they are a wizard...?
- Can they wear armour despite
being a wizard? Yes, the FAQ clarifies this.
- Can they buy arcane items? Yes, No, Maybe? If No, why aren't they a Wizard in this specific instance but in all the others?
If you say "Yes" to one of the above, you have to say Yes to all of them. If you say No to one of the above, you have to say No to all of them (except the armour one as the FAQ is clear on that part, and of course they can cast spells) - otherwise you are yourself picking and choosing what benefits to get, and thats not how it works.
My suggestion is to don't get riled up about this and sort it out within your own gaminggroup (but honestly, how often will this become an issue?), because this won't get FAQ:ed soon, if ever.
Personally I'm in the "Yes, they can buy Arcane items"-camp (simply because there are less contradictions), but its not a hill I'm willing to die on in a forum debate, because I'll probably never play a fighty character with wizarding hat + arcane item combo to begin with.