The battery is 3 amp-hour, not "3Amp". Amp-hours is a capacity rating meaning the battery can deliver approximately 3A for 1hr, or 1A for 3hrs, etc. The device drawing current from the battery is what determines the amps in the circuit.

The slip ring may not take the motor load, but should easily handle all the other functions like lights, recoil, elevation, turret rotation, etc.

I have a 12 wire slip ring that works fine (220V, 2A), but I don't have the motor wires going through it. If you have enough wires in the slip ring (some have up to 24 wires I think), you can use two or three of the slip-ring wires grouped together for each motor lead. They will "share" the current load and I'm guessing that would be enough to handle the motor current draw.