Do you have a spunding valve?
I have not done this before, but my first thought would be put spunding valve on gas post of receiving keg set to same pressure as current kegs. Then run liquid post to liquid post and then attached CO2 to gas post of emptying keg (set at same pressure as everything else). Then just very lightly loosen the spunding valve so that gas very slowly gets out (as filling keg very slowly fills).
If you don't have a spunding valve, I would just disconnect both kegs, pull the pressure release pins let the pressure out, wait an hour, all the pins again, wait another hour, attach liquid post to liquid post, attach CO2 tank to emptying keg, and then have a gas out connector with a blow off tube to attach to the receiving keg. Then just slowly turn the CO2 on to begin a slow transfer.