I'm sorry if I'm dense, but if you only have three teams with cap problems, you really only have three teams to "cut in 18 hours." That seems doable.
The other fifteen should be cuttable to within one or two. Many of the players are the same.
I have fifteen teams, nine of which are trad, and two of which are close to the cap. I will have one or two bubble players on some of the seven, but probably only have to really worry about the two teams that might be close.

The trade-off is further delaying the season.
What have you been doing for the last three months?
Dave N