Transmission oil

I would flush the trans without hooking the cooler lines up to the new radiator, disconnect from the old radiator the flush the lines and trans itself. Flushing involves pushing new trans fluid through the transmission and lines through as it disperses the old. You will use extra fluid doing this. The trans was compromised with the coolant as it effects the clutches, so hope you don't have any future issues Then replace the radiator, hook up the cooling lines to it, and install new fluid. Install new coolant, bleed system and be on your way.