Expenses for the DC should come out of shared money, and each partner should have access to their own money on top. The amounts depend on what is affordable and priorities.
Clothes for growing DC are essential. Clothes for DC who already have plenty and just because you like them, not so much. Days out for DC are nice to have, but need to be affordable.
Each partner should be able to buy their own clothes, tech etc, but this might be lower priority if the budget is tight.
Ideally everything joint, including for DC should come out of the joint pot and personal come out of a personal pot. Separate bills from day to day spending, and don't touch the bills account, except to check each month that there's enough money in for the next month's direct debits. Send some money on pay day to a difficult to access savings account for annual and irregular expenses like car repairs, Christmas etc.
Does the person with the bipolar disorder have the discipline to run 2 accounts - perhaps have 2 cards in different colour, so if you're buying for family/DC, pay with the blue card, if buying for self, pay with the red card - I'm not sure if this is practical though?