Yes, the fourth chronicle sheet awards the PC with three experience points and four prestige points at either 2nd, 4th, or 6th level with the corresponding amount of gold. Thus a PC with these four chronicle sheets can advance four levels.
The PFS rules are merely a subset of all the rules. You can play the module in campaign mode with PFS characters if you like - or not. As GM you set the limits when playing in campaign mode. For example, when I ran this for a group of four players I limited the players to a 15-point buy and only allowed use of Paizo books published before the release of the module with the exception of the Dragon Slayers Handbook.
Furthermore, it is entirely conceivable to play the module with a PFS-legal character in campaign mode, get the four chronicle sheets and apply them to an identical character - but one that has a PFS character number! Granted the PFS character may not be at the same level as the PC that completed the module, but the player will have had the enjoyment of playing their "PFS character" outside of PFS play for several levels and can make any adjustments they see fit as they apply these chronicle sheets to their PFS character.
Check out paragraph 2 under the section entitled "Receiving Credit."
From the sound of your post, I assume you are only running the sanctioned portions of the module. If so, the fourth chronicle sheet does not apply. The fourth chronicle sheet is only awarded if you play the entire module as detailed in paragraph two of the section entitled "Legal Pathfinder Society Characters."