From a chief pilot's perspective, you should let him know. That is if you are not wanting to burn any bridges. As you said, you can still fly as SIC and earn your paycheck until you leave for the airlines. If the offer goes away you can attend recurrent at a later date.
Rogersmith made the comment that it's already factored in the budget, but I'm here to tell you that may not be the case. Yes the recurrent is factored in but what about the cost of recruiting, integrating and possibly sending the new hire to an initial. Depending upon the stability of your current flight department, that may not have been factored in. I see to many guys that think the world owes them something. When costs go up the bean counters start looking at places to cut. The flight department is one of the first places they look. Also think about the older guys in the flight department that may not have an airline offer or are too old to start one. Your actions could shutter a flight department and put these guys on the street. Put yourself in the Chief Pilot's/Director of Aviation's shoes, wouldn't you want to know?