But if you then went to bed, you would suffer the cost at the end of your sleep. I do understand your logic of saying the cost fuels the sceptre, I will have a think about that as it might change my approach .
But regardless how you see the sceptre, as something that just hurt you or something that gets its power from hurting you, it hurts you. It probably hurt the wielder over time, not just at one point, but we are only talking about one point of Agility and one point of Wits.
By deducted them at the end of the Quarter Day means there is no real downside. You can use the sceptre for five hours, put the sceptre in a chest and then at the end of the Quarter Day, no one is there to take the damage, even though someone held the sceptre for five of the six hours of the Quarter Day or if it doesn't matter who held the sceptre during the Quarter Day, only who has it at the end, then the players can just give it to someone that isn't bothered with damage or better, giving it to someone that is already broken in Agility or Wits. I mean, you can't get more broken.
This is the reason my players pay the price in the beginning of the Quarter Day. Easy to keep track and I don't have to deal with all those exploits.
I should say that there is no official answer (that I know of) of this, only opinions.