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You don't say which FX body you have so the answer is not straightforward.
If you have a D750, then you are correct in saying the perceived FL will be 450mm using a 300mm lens. OTOH, if you have a D800, then the perceived FL will be substantially less because the D800 starts off with more (36) MPix than the D7200.
The other factor is the Aperture of the lens. If the 300 is the long end of a zoom lens and say f/5.6, unless you have really good light, you're probably going to run into problems with noisy files when using the D7200.
If your FX camera is 24Mpix and you have a 300/2.8 then I would probably go for the crop camera.
You don't say which FX body you have so the answer... (
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Huh? The "perceived focal length" would be exactly the same whether on a D750 or a D800.... both are FX cameras, so on both a 300mm will behave as a 300mm on full frame.
Apertures don't change with sensor size, either. So, not sure I understand your next point, either. f5.6 is f5.6, whether it's on a D750 or a D7200. I think you mean that the D7200 will have more of a limit on high ISO usage, due to the smaller sensor... or to put it another way, higher ISOs will be more noise free on the FX camera.
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Why not just add a TC14E III at a fraction of the cost. You will probably see less IQ loss than either alternative.
A DX camera will give higher effective magnification (1.5X vs 1.4X)
and will out-resolve the lens + teleconverter.
Either can be expected to be better than switching to DX format on an FX camera. (Which is the identical to later cropping the image.)
A 24MP D750 in DX mode makes it a 10MP camera.
A 36MP D800 in DX mode makes it a 15MP camera.
While both those are respectable, 24MP from an actual DX camera such as the D7200 could be expected to give better overall image quality.
The alternative, getting a longer lens to use on the FX camera, would give maximum image quality. But, it depends upon what you do with your images, too. Unless you print big... really big... or make some other high use of the images, you are likely to be the only person who sees the difference, when looking at your original files at ridiculously high magnifications like 100% on your computer monitor. To get that ultra high quality image, you'll not only need to spend a lot of money, you'll also have to haul around a lot larger and heavier gear... probably a tripod to sit it on too.
So, all things considered, generally your best bet would be to get a DX camera and use it with current lenses. FX and DX cameras can complement each other very well.