Fundamentally you're on the right track; with the camera at 5200K and the lights really pumping out 5000K, the pictures will have a warm cast to them, but this can be easily corrected in post processing.
Most important though is to make sure that all the lights that you're using have indeed the same colour temperature. And, given that they're fluorescent, have a suitable spectrum.
You can't really colour correct a spectrum that has two or more significant bumps at very different wavelengths. In that case you'd always end up with some colours off.
The whole idea of colour temperature is related to black body thermal radiation. Essentially it's comparable to what you get when you heat a piece of iron. The hotter it gets the whiter it radiates because more and more shorter wavelengths of radiation start to contribute. But all the time, the spectrum starts with the highest output at long wavelengths and tapers off towards shorter ones.
The white balance of the camera basically reshapes the spectrum from one black body radiation shape to another, and is only able to do so if the shapes are basically the same form..
Unless the lamps are intended for photographical use, you may have to try one out before jumping in and/or spending $$ on them.