Shinano would had carried 3x 2.774ton triple 46cm turrets, 2x triple 190ton 15,5cm turrets and 6 twin 34,5ton 10cm turrets totalling 8.909 say 9.000tons of armament plus ammo and barbette armour, I doubt she would had any issues with stability even with a double hanger deck layout.
The problem is the Japanese wanted carriers as fast as they could get them and creating another hanger deck would cause a later construction date, more materials, and design workhours for calculations.
Stability may or may not have been an issue. It is not simply a case of the weight but the height at which it is carried.
As a battleship much of the weight of the armour, armament, munitions etc is placed low in the ship. In a carrier with an armoured flight deck like Shinano much more of it is carried much higher in the ship.
It is also worth noting that Shinano had her main belt side armour halved in thickness during conversion. And yet as completed her full load displacement remained about the same as the Yamato class battleships (c72,000 tons full load). Her draught was also about a foot greater than that of the battleships. So despite losing all the kit mentioned a lot of other stuff including the armoured flight deck, aircraft ordnance, aviation fuel etc had to go in.
The practice in the armoured carriers of Britain, the USA and Japan in the case of Taiho, was that the flight deck armour covered the hangar area between the lifts. So with a hangar 164m long and with a width of 18-34m (from the drawing in an earlier post most of it looks nearer the latter than the former) that is an area of somewhere between 2,952-5576 square metres of flight deck to be armoured. Shinano had 3" of armour on her flight deck but that was AIUI laid on a backing of 1" of ordinary steel as in the Taiho. So that is a total of 4" of metal to be supported at some height above the main deck. At a rough estimate that is somewhere between 2,300-4,300 tons of metal depending on the hangar width being covered. Given my above comments about hangar width probably nearer the latter.
Then the question of hangar height. The hangars in Taiho were designed to operate the same types of aircraft as Shinano, had a clear height of 5m. That means that the c4,000 tons of flight deck in Shinano was carried 5m + the depth of the substantial beams to support its weight above the battleship upper deck.
Add a second hangar into the ship and you have to carry that c4,000 tons of armour another 5+ metres higher still. That has to have some effect on her overall stability.
Whether that extra hangar would be enough to destabilse her altogether I don't know as I'm not a naval architect. But it demonstrates that the matter is not as simple as looking at the weights.