I have two options here - ask you to re-read my previous post or cut & paste it.
As far as I know, the Sidekick Sport was a North American market vehicle, released to market at a point in time when Suzuki was not fitting immobilizers to vehicles for that market, and the immobilizers traditionally used did not immobilize by shutting down the fuel pump.
We can assume that you have an optional Suzuki "anti theft" device which would probably be more of a remote lock alarm type device, or we can assume that you have a third party "anti theft" device that does shut down the fuel pump (not much of a deterrent since you have apparently already figured out how to bypass it) - the point here being that making assumptions gets you no closer to identifying the problem and that's the first step into resolving it.
Stalling at any speed after driving a few miles could have any number of causes, lack of spark, loss of power on the ignition system, lack of fuel, loss of power on the fuel pump or injectors, loss of input from crank/cam sensors, defective ECU, and the list goes on.
Does it restart easily when it stalls? If not, that's where & when the troubleshooting starts - check for spark, check for fuel.
By the way - IF you do have an OE Suzuki immobilizer and it is shutting the engine down, I'd expect the dash lights to tell you that - hence my question on the lights.