I don't think this is as much an issue as you might think. The reason is that the players on your team will probably be better than the people available in the free agent market. If they're not, then you're probably not winning anyway.
So let's say you get to the 5th day in the week (Friday) and realize you're down by 2 wins, and all your starters won't be starting on the weekend. Do you drop 4 of your pitchers to pick up 4 pitchers that will be starting? I doubt it, since the other teams in the league would jump all over the people you dropped.
And if you're planning on rotating 2 pitching "slots" by droping a person each day, the quality of pitcher you'd be adding would be atrocious. Think people like Brett Tomko (5.28 ERA, 1.52 WHIP, 13 wins), Jason Johnson (4.18 ERA, 1.56 WHIP, 10 wins) and so on. Yes, you MIGHT pick up a win or 3-4 extra Ks, but you'll be losing in WHIP and ERA. Plus, if you don't get the win, you're going to be tanking in 4 of the 5 pitching categories.
Basically you're suggesting fielding a large pitching staff made up of players that nobody else wants on their team...