So what caliber (.Inch) would you recommend for a battlecruiser?
Technically speaking, there were plans with Battlecruisers armed up to 508mm (20-inch) guns! (Lord Fisher's HMS Incomparable) but the were honestly a bit 'silly'. The largest battlecruisers actually started where the US Lexingtons and the Japanese Amagis, wich were both to be fitted with 406 and 409mm (16 and 16.1-inch) guns.
The main deicing factor for the armament is just when your battlecruiser is laid down. 1906-1910 I'd say 305mm. 1911-1915 356mm. 1916-1920 381mm. 1920-1922 406mm guns.
edit: one thing, at first glance your ship has a draft (the height of the hull below the waterline) of some 15 meters, if not more. You'd hardly need anything over 10-11m.