Hi, and welcome to the forums.
My dear friend, what you have there is a toy. If your intention is just to make it go again and have a little fun with it till it breaks, well, that's about all you can expect, really. No replacement parts for them. That's the difference between a Toy and a hobby-grade RC.
That Tenergy battery would be fine, providing its dimensions will fit the battery bay's dimensions. The bigger the milli-amp-hour (mah) number, the longer a pack will last compared to one that has a smaller number.
As for what will last longer, the car battery or the remote battery.....definitely the remote batteries. They don't get as much drain as the car's battery. So no worries there.
Replacing your remote won't get you spins. You need power to do that. (WHY is that a priority anyway? Not exactly the most fun.) And I doubt that Radio Shack buggy has much of a motor.
If all you want out of an RC is giggles watching it do doughnuts, then yeah, get the other Radio Shack car. But if you'd like to be able to replace parts that break, you really should look into hobby grade stuff. Tamiya makes a Wheely-King and a few other models that are hobby grade, but made for shenanigans like you're looking for.