The Professor does not have any secret knowledge. Looking at The Professors rules in the linked PDF there's no indication that the Storyteller would ever tell The Professor when they wake up in the night if someone has died that same night.
The Professor only knows who is dead when Dawn arrives and the Storyteller lets the group know who, if anyone, has died. That's how The Professor gets their knowledge in the game. One of the last lines of The Professor examples has:
At dawn, after declaring which players died, declare which player is alive again.
Let's work with the example in the question:
Unbeknownst to the Professor, that player was regurgitated by the Shabaloth and is now alive. The Storyteller shakes their head no, because the Professor must choose a dead player.
The Shabaloth eats 2 players every night and may, the following night, regurgitate one of the two players. In the given example The Shabaloth eats 2 players and the following Dawn The Storyteller shares with the group that 2 players are dead.
This gives The Professor information on who the 2 dead players are, one of which they (The Professor) may resurrect the following night. Unfortunately, in the given example The Shabaloth goes before The Professor, regurgitating one of the players which changes their status to "alive". When it is The Professors turn they coincidentally choose the regurgitated player and the Storyteller must inform The Professor that the chosen player is not dead by giving them a head shake:
The Storyteller shakes their head no, because the Professor must choose a dead player.
In this instance, The Professor does get the knowledge of a player coming back to life specifically because:
- The Professor can only choose dead players.
- The player was dead the previous night and would be up for consideration.
The rules state in multiple places that The Professor must choose a dead player specifically. This means they have prior knowledge of who is considered dead players, even if that knowledge changes in the same night they resurrect.
I would consider the above example an edge case. Otherwise, The Professor could have a lucky guess one night and resurrect a dead player the same night they died which the rules do not touch on at all.
The above is my interpretation of the linked rules. The rules will not cover every edge case. When in doubt, The Storyteller will always have the final say.