So I ordered that USB cable from eBay and it works very well! It automatically puts the phone into Dock Mode when plugged in.
I have the phone set to launch CarHome Ultra when it goes into Dock Mode, and CarHome Ultra actually turns on Car Mode automatically when it launches. It can also be set to turn on the phone's Drive Mode if you want, and it turns on Bluetooth, and turns off WiFi also.
So now I just plug the phone into the USB charger, it launches CarHome Ultra automatically, turns on Car Mode and Bluetooth which then pairs with my car's bluetooth. I even have the home button working so that it goes back to CarHome Ultra when pressing the home button (using Xposed framework with EnableCarDock module on stock, rooted odexed Sprint S4).
Too bad my car's bluetooth is older and does not have the media playing profile so I can't stream the phone's sound via bluetooth. I'd either have to use the headphone out to the car's auxillary (works great but bluetooth phone can't be used when the headphone jack is in use - stupid Samsung!), or buy another bluetooth to auxillary device just for music/sound streaming, or buy iBolt's Y-cable (which many people seem to report poor sound quality or lots of noise).