I think it is unlikely that cutting the side flanges of will cause any warping. These vises are ductile cast iron which is heat treated after casting and is very unlikely to have any internal stresses that would be relieved by machining off the flanges.
However, it might not be a good idea, not because the vise is weakened but because the bolts clamping on the side flange add to the stiffness of the vise when it is closed. Because the jaws exert a force above the vise base when it is closed the tendency is for the vise to arch its back. I have never tried it but I think if you tightly closed an un-mounted vise you would be able to measure the bending on the bottom surface.
With your proposed clamping at each end you might even find the vise distorts enough to lift the middle portion clear of the table. This would tend to turn the vise into a kind of drumhead and I would expect to get significant vibration when taking heavy cuts.
You should try and get more information on the Leave vise to see if it has holes for hold-down bolts located under the center jaw. I have the Kurt double-lock vises and these do have holes in the base that allow them to be bolted down at the center when side clamping is not possible. I do not know if Kurt makes a small version of the double-lock.