I'm very confident after some more testing that it's probably possible to get 4 influence by day 12, but it requires such sacrifices that you will absolutely destroy your discipline and not be able to implement any new policies, and so doing this will have no value. Assuming that you can at the earliest get these rules changes passed on day 19 then you can at the earliest sound out the staff's positions on you dealing out corporal punishment on day 26 and pass it on day 33. Is it possible to get the necessary quests done by day 33?