You have a bad regulator. A properly functioning one will maintain close to the same out side pressure over a wide range of in side pressures. Not only is it malfunctioning but in a very strange way. Apparently as the inlet pressure drops (as the CO2 liquid cools its saturated vapor pressure decreases) the outlet pressure is increasing. In a properly functioning regulator the outlet pressure drops slightly as inlet pressure drops but it's by such a small amount you don't notice it. I can't explain how it might be causing what you see. They are cheap enough. Toss it.
As to OP's question: it doens't matter one bit whether you apply the CO2 before the beer is cool or not. As the beer cools and the solubility of CO2 increases more will dissolve from the head space causing a reduction in head space pressure which a poperly functioning regulator will sense and allow more gas to flow from the bottle thus restoring head space pressure to close to the regulator setting.
It is, of course, a good idea to apply the gas immediately in order to purge the headpace of any air and an even better idea to fill against CO2 counter pressure which CO2 got into the keg by displacing steam or water so that there is no O2 to displace. If you are going to set and leave things alone set
P = (V + 0.0033)/(0.01821 + 0.090115*exp(-(T-32)/43.11)) - 14.695
for the volumes (V) you want at the serving temperature (T, °F). If you wish to accelerate the process of carbonation you can set a higher pressure initially but don't forget to back it off after a couple of days.