Is it possible to replace this radar gun handle wiring with a battery?
The answer is yes.
The circuit inside the handle would be a buck converter.
Going by the available information on the Stalker Radar Gun and its accessories, 2 battery configurations appear feasible:

Six 1.2V 2500mAh rechargeable NiMH batteries in series
Two 3.6V 2500mAh rechargeable Li-ion batteries in series.

They would be capable of 2 hours of continuous transmitting after a full recharge.