Ok it's the fuel sending unit. Gimme a sec lemme see what I can dig up.
Well I haven't found any TSBs on it. Most were replaced under warranty. Without an intact warranty you can ask your service advisor to contact Infiniti NA for a goodwill repair. Alternatively you can call Infiniti NA yourself and voice your concerns and see what they can do for you.
Only thing I found on the governments site (one which anyone can search at odi-nhtsa.dot.gov) is the fuel filler hose being cracked which would result in fuel spillage while refueling, which doesn't relate to your problem.
Make: INFINITI Model: FX45Model Year: 2005Manufacturer: NISSAN NORTH AMERICA, INC.Mfr's Report Date: NOV 28, 2005NHTSA CAMPAIGN ID Number: 05V555000 N/A
NHTSA Action Number: N/A Component: FUEL SYSTEM, GASOLINEPotential Number of Units Affected: 192659Summary:
ON CERTAIN PASSENGER AND SPORT UTILITY VEHICLES, THE FUEL FILLER HOSE MAY CRACK AND RESULT IN FUEL LEAKAGE FROM THE HOSE WHILE REFUELING.Consequence:
FUEL LEAKAGE IN THE PRESENCE OF AN IGNITION SOURCE COULD RESULT IN A FIRE.Remedy:
DEALERS WILL REPLACE THE FUEL FILLER HOSE. RECALL BEGAN ON FEBRUARY 27, 2006, FOR NISSAN OWNERS, AND ON MARCH 13, 2006, FOR INFINITI OWNERS. OWNERS MAY CONTACT NISSAN AT 1-800-647-7261 OR INFINITI AT 1-800-662-6200.Notes: