TLDR : The odds are that the Fidelius charm would serve its purpose depending on how exactly it works or what happens technically that it is used to safe-guard a secret.
Full explanation-
Now, in my understanding, there are 2 cases that may arise here:
For ease of understanding, I'll be calling all those (combined) from whom the secret is to be kept as X.
- Case 1: Secret is - A location’s address which means that the entire place is to remain hidden from X
- Case 2: Secret is - The presence of a particular person(s) at a location from X and NOT the entire location's address or existence per se
As explained by Filius Flitwick in PoA
"As long as the Secret Keeper refused to speak, You-Know-Who could search the village where Lily and James were staying for years and never find them, not even if he had his nose pressed against their sitting room window!"
Now other than this, IIRC, there is no other canonical explanation of what exactly is happening in a Fidelius charm. IMO, there are 2 options:
- Option 1: The charm works because X is simply not able to see some or all part of the objects included in the secret even when standing right in front of them
- Option 2: Under the charm's effect, X's brain isn't able to process all the information correctly even when standing right in front of the secret and looking at everything. Because of this X is unable to act accordingly which eventually also serves the main purpose of keeping the secret.
Combining the 2 cases (of what the secret is) and 2 options (of how the charm works), 4 possibilities arise (Now, I'm assuming that if X is taking the exact place's name while using the Floo-network without being aware of its intended secrecy from itself, then it is doing so, either by chance or it knows the location per se but is unaware or unsure of it being used for hiding sth that X seeks and just to try its luck, plans to go there using the Floo network to search what it is looking for) -
Possibility 1: Location's address is a secret and the charm makes it invisible to X
Floo-network effect: Either, the charm stops X's entry making it getting lost in the network unable to reach the hidden place or if the charm only keeps the outer facade invisible then X's "long shot in the dark" works and X makes a serendipitous discovery of a weakness in the Fidelius charm and puts it to its use.
Possibility 2: Location's address is a secret and the charm affects X's brain to know why it wants to visit this place
Floo-network effect: X gets inside the location and is not able to comprehend why it is there, gets confused and walks out.
Possibility 3: Location's address isn't a secret per se but the presence of someone in particular is and the charm makes that person invisible to X
Floo-network effect: X gets inside the location and is not able to see the person it is looking for, disappointed and failed X comes out.
Possibility 4: Location's address isn't a secret per se but the presence of someone in particular is and the charm affects X's brain to know why it is searching for that person.
Floo-network effect: X gets inside the location, sees that person and is not able to process all the information pieces (either because it forgets its purpose or forgets the face of the person it is looking for) and comes out confused.