The items are still stored in the sylladex and will work as expected, except they are not rendered nor clickable (only for the tree modus). The card also cannot be removed by right-clicking it.

The empty card in the picture is a stack of 128 chests

The problem is more so that 128+ stack sizes aren't supported by the sylladex in general (specifically in serialization, where the method used limits the stack size to fit a java byte).

However, since the sylladex is meant to adhere to normal stack size limits at the moment, the more relevant problem is how you ended up with those large stack sizes in the first place.

I used Nether Chested, which adds a nether chest that can hold 8 times the stack limit in a slot.

Alright. The solution that I want to suggest then is that I make captchaloguing take out one regular-sized stack from the captchalogued inventory slot instead of the whole stack in the slot.

The bug can still occur when holding the stack in hand instead of hovering on a slot and captchaloguing.

Seems about right. Captchaloguing a container slot and captchaloguing the held item happens in two different functions. But while a modded container could be changed to hold oversized stacks, I wonder how one might hold an oversized stack in the player inventory.