I never bother using the outlet to carbonate. The CO2 gets in eventually.
If I am in a hurry, I use a higher pressure than desired (eg. 30 PSI in a beer at 40F that I plan to serve with 12 PSI), put the keg on it's side and rock it like a baby in my arms for 10 minutes or so. You will hear/feel gas running into the keg as you agitate. I always keep the air inlet valve up to avoid backflow into the air line. Caution advised b/c you may overcarb. Frequent sampling during the carbing process will help avoid this adverse event
If I am in a hurry, I use a higher pressure than desired (eg. 30 PSI in a beer at 40F that I plan to serve with 12 PSI), put the keg on it's side and rock it like a baby in my arms for 10 minutes or so. You will hear/feel gas running into the keg as you agitate. I always keep the air inlet valve up to avoid backflow into the air line. Caution advised b/c you may overcarb. Frequent sampling during the carbing process will help avoid this adverse event