It won't work as is- because of length. But you can possibly make it work by adding a spacer (thick washer) around the cartridge base once you place it in the shell holder before running up into the FCD. Measure the difference between the HH case length, which is what the Lee FCD is designed for, and the Flanged case length. That difference determines the approx thickness a spacer will have to be so it will work. I'm assuming you have a 375 Flanged Magnum? The book difference shows a standard Flanged Magnum case to be about .1" longer than the Belted Magnum. If that is correct for yours, the spacer (washer) will need to be about .1" thick. Your pic seems to show the difference to be more than .1"... but that could just be a photographic distortion?
I use a 458 Lott FCD for my 450 Watts which is a little longer than the Lott. I simply add the correct thickness spacer (an aluminum washer) over the cartridge and run it up into the FCD to crimp. Pic below shows that set up
The FCD is a collet die with 4 fingers which compress inward on the rim of the case mouth as the shell holder pushes up against the bottom of the sliding collet body that is within the die body. Very simple once you see how it works. The Lee FCD, IMO, makes a superior crimp. And no more expensive than the FCD is, it would be worth it to give it a try with a spacer (washer).
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