I would think as long as the (ahr), ampere hour rating, is equal to or larger than the rechargeable battery you are replacing.
Sure, the device, load, connected to the battery will only draw the amount of current, amps, that it needs to operate at.
I would recommend a fuse be put in series between the battery and the load for short circuit protection. If you know how many watts is comsumed in the circuit in the cable you can find the amp rating needed for the fuse. watts = volts x amps. watts/volts = amps. Just double the size of the fuse rating for the two cables. If you do not have any values, just pick a number,say 1 amp. Install the fuse at the battery, positive terminal. You should be able to pick up an in-line fuse holder with leads on each end and a fuse at any automotive store.
One thing I would check first is the voltage at the output terminals of the car battery. Depending on the charge in the battery the DC voltage could be as high as 14 volts.
How critical is the amount of voltage that can be connected to the cables? Is there a tolerance given, + or -?
Also make sure you use a sealed car battery. The type with vent caps give off hydrogen gas fumes......