A core is just nuclear fuel with enough other parts that it can be handled. Of course there needs to be a control mechanism for adjusting the output, you just can't use that for eliminating heat output. Once the core has been turned on there is decaying fission products producing heat, that is why you can't turn off a nuclear power plant in an instant, and why you might want to get rid of the core. Unused cores can be stored with few special precautions as the unspent fuel decay too slowly to be a major hazard.
The ship would carry one small nuclear reactor, and several tons of coolant/propellant. The only way the cooling system is actually going to keep up is by ejecting the hot coolant into space, which is also what creates thrust. You can't recycle the coolant as there is no efficient way to cool the coolant.
In effect, if you don't want to dispose of the core you have to keep the engine on consuming coolant/propellant, depending on a lot of factors that may or may not be viable.