What JA said!
I wouldn't think that you would need much head space, as long as the surface of the beer is exposed the C02. Pressure is exerted equally across the surface. For instance when the keg is 3/4 empty the carbonation level does not increase. I do the following to speed up the process. Some rock and roll the keg during a similar process.
Once I transfer in through the dip tube, I then hook up the gas to that same side. With the regulator set at about 5#, the little ball valve after the regulator closed, and the relief closed I gradually open the little ball valve after the regulator and let the C02 in slowly, and let the pressure equalize. I then turn off the little ball valve, slowly release the relief valve, and repeat. I do this 4 or 5 times. I use to let the gas rush in, but it would froth up and make a mess when I released the pressure. After this process I set it at 25# and leave it for 1-2 days, then it is ready to serve.