I've become more confident in my wiring the last couple of builds, but since this build is for someone else I want to be sure this is correct.  Can someone please verify? Not sure if the current sensor is necessary but if so I'd wire the one from the ESC to the one on the FC, if needed.

   

That is correct apart from one observation. You have the 5V wire from the VTX going to a BAT pad which will be LiPo voltage. If your VTX can only take 5V then connecting it to the BAT pad will fry the VTX. If the VTX can take a large input voltage range which is above the LiPo being used then that is fine.

That was my error - it takes 7-24V.  

This is the VTX.  Now that I think about it, that's probably why my camera got smoked and the VTX didn't on my other FC (that all 5V pads were reading 14V on) - The VTX could handle it but the camera was 5.5V.

   

So is there a difference between VBAT and BAT?

There is technically not difference between BAT and VBAT. Occasionally, sometimes the vbat gets filtered. Usually not.