The bead will vanish after its hour is up
The full text of Wristpocket is:
You flick your wrist, causing one object in your hand
to vanish. The object, which only you can be holding
and can weigh no more than 5 pounds, is transported
to an extradimensional space, where it remains for
Until the spell ends, you can use your action to summon
the object to your free hand, and you can use your
action to return the object to the extradimensional
space. An object still in the pocket plane when the spell
ends appears in your space, at your feet.
I do not read "where it remains for the duration" to overrule the effects of time on the object, in a kind of magical preservation stasis. It just means the item is now that space until the end of the duration, at which point it appears in your space, at your feet. You even can use your actions to get it out and put it back during the duration, in contradiction to "where it remains". The spell says nothing about preservation.
If, for example, you put an egg in there that is just short of hatching, it will hatch in the extradimensional space. The space is not somehow conserving it unchanged or stopping the flow of time for it, or it would need to say so. Likewise, if you put an object there that will expire, like the bead, it will expire. The space is not conserving it or stopping the flow of time for it either. It is just that whatever happens to the object, happens to it in that space. The bead never leaves the space to somewhere else when it vanishes, it ceases to exist.
(I don't think this is a case of specific over general -- both rules are specific.)